Research Article Analysis of Functionally Graded Material ...orca.cf.ac.uk/92599/1/247932.pdf · Research Article Analysis of Functionally Graded Material Plates Using Triangular

Research ArticleAnalysis of Functionally Graded Material PlatesUsing Triangular Elements with Cell-Based SmoothedDiscrete Shear Gap Method

S Natarajan1 A J M Ferreira23 S Bordas4 E Carrera35

M Cinefra5 and A M Zenkour36

1 School of Civil amp Environmental Engineering University of New South Wales Sydney Australia2 Faculdade de Engenharia da Universidade do Porto Porto Portugal3 Department of Mathematics Faculty of Science King Abdulaziz University PO Box 80203 Jeddah 21589 Saudi Arabia4 Institute of Mechanics and Advanced Materials Cardiff School of Engineering Cardiff University UK5Department of Aeronautics and Aerospace Engineering Politecnico di Torino Italy6Department of Mathematics Faculty of Science Kafrelsheikh University Kafr El-Sheikh 33516 Egypt

Correspondence should be addressed to S Natarajan sundararajannatarajangmailcom

Received 25 September 2013 Revised 2 January 2014 Accepted 16 January 2014 Published 8 April 2014

Academic Editor Hung Nguyen-Xuan

Copyright copy 2014 S Natarajan et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

A cell-based smoothed finite element method with discrete shear gap technique is employed to study the static bending freevibration and mechanical and thermal buckling behaviour of functionally graded material (FGM) plates The plate kinematicsis based on the first-order shear deformation theory and the shear locking is suppressed by the discrete shear gap method Theshear correction factors are evaluated by employing the energy equivalence principle The material property is assumed to betemperature dependent and graded only in the thickness directionThe effective properties are computed by using theMori-Tanakahomogenization method The accuracy of the present formulation is validated against available solutions A systematic parametricstudy is carried out to examine the influence of the gradient index the plate aspect ratio skewness of the plate and the boundaryconditions on the global response of the FGM plates The effect of a centrally located circular cutout on the global response is alsostudied

1 Introduction

With the rapid advancement of engineering there is anincreasing demand for new materials which suit the harshworking environment without losing their mechanical ther-mal or electrical properties Engineered materials such asthe composite materials are used due to their excellentstrength-to-weight and stiffness-to-weight ratios and theirpossibility of tailoring the properties in optimizing theirstructural response But due to the abrupt change in materialproperties from matrix to fibre and between the layers thesematerials suffer from premature failure or from the decayin the stiffness characteristics because of delaminations andchemically unstable matrix and lamina adhesives On thecontrary another class of materials called the functionally

gradedmaterials (FGMs) aremade up ofmixture of ceramicsand metals and are characterized by smooth and continuoustransition in properties from one surface to another [1] As aresult FGMs are preferred over the laminated composites forstructural integrity The FGMs combine the best propertiesof the ceramics and the metals and this has attracted theresearchers to study the characteristics of such structures

Background The tunable thermomechanical property ofthe FGM has attracted researchers to study the static andthe dynamic behaviour of structures made of FGM undermechanical [2ndash5] and thermal loading [6ndash12] Praveen andReddy [7] and Reddy and Chin [13] studied the thermoelasticresponse of ceramic-metal plates using first-order sheardeformation theory coupled with the 3D heat conduction

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 247932 13 pageshttpdxdoiorg1011552014247932

2 Mathematical Problems in Engineering

equation Their study concluded that the structures madeup of FGM with ceramic rich side exposed to elevatedtemperatures are susceptible to buckling due to the through-thickness temperature variation The buckling of skewedFGM plates under mechanical and thermal loads was studied[9 14] by employing the first-order shear deformation theoryand by using the shear flexible quadrilateral element Effortshave also been made to study the mechanical behaviour ofFGM plates with geometrical imperfection [15] Saji et al[16] have studied thermal buckling of FGM plates withmaterial properties dependent on both the composition andtemperature They found that the critical buckling temper-atures decreased when material properties are consideredto be a function of temperature as compared to the resultsobtained where the material properties are assumed to beindependent of temperature Ganapathi and Prakash [9] havestudied the buckling of FGM skewed plate under thermalloading Efforts have also been made to study the mechanicalbehaviour of FGM plates with geometrical imperfection [15]More recently refinedmodels have been adopted to study thecharacteristics of FGM structures [17ndash19]

Existing approaches in the literature to study plate andshell structures made up of FGMs use finite element method(FEM) based on Lagrange basis functions [14] and meshfreemethods [20 21] and recently Valizadeh et al [22] usednonuniform rational B-splines based FEM to study thestatic and dynamic characteristics of FGM plates in thermalenvironment Loc et al [23] employed isogeometric finiteelement method to study thermal buckling of functionallygraded plates Even with these different approaches theplate elements suffer from shear locking phenomenon anddifferent techniques were proposed to alleviate the shearlocking phenomenon Another set of methods have emergedto address the shear locking in the FEM By incorporating thestrain smoothing technique into the finite element method(FEM) Liu et al [24] have formulated a series of smoothedfinite element methods (SFEM) named as cell-based SFEM(CS-FEM) [25 26] node-based SFEM [27] edge-basedSFEM [28] face-based SFEM [29] and 120572-FEM [30] Andrecently edge based imbricate finite element method (EI-FEM) was proposed in [31] that shares common features withthe ES-FEM As the SFEM can be recast within a Hellinger-Reissner variational principle suitable choices of the assumedstraingradient space provide stable solutions Depending onthe number and geometry of the subcells used a spectrumof methods exhibiting a spectrum of properties is obtainedInterested readers are referred to the literature [24 25] andreferences therein Nguyen-Xuan et al [32] employed CS-FEM forMindlin-Reissner platesThe curvature at each pointis obtained by a nonlocal approximation via a smoothingfunction From the numerical studies presented it was con-cluded that the CS-FEM technique is robust computationallyinexpensive free of locking andmost importantly insensitiveto mesh distortions The SFEM was extended to variousproblems such as shells [33] heat transfer [34] fracturemechanics [35] and structural acoustics [36] among othersIn [37] CS-FEM has been combined with the extended FEMto address problems involving discontinuities The above listis no way comprehensive and interested readers are referred

to the literature and references therein and to a recent reviewpaper by Jha et al [38] on FGM plates

Approach In this paper we study the static and the dynamiccharacteristics of FGM plates by using a cell-based smoothedfinite element method with discrete shear gap technique [39]Three-noded triangular element is employed in this studyThe effect of different parameters namely the material gra-dient index the plate aspect ratio the plate slenderness ratioand the boundary condition on the global response of FGMplates is numerically studied The effect of centrally locatedcircular cutout is also studied The present work focuses onthe computational aspects of the governing equations andhence the attention has been restricted to Mindlin-Reissnerplate theory It is noted that the extension to higher ordertheories is possible

Outline The paper is organized as follows the next sectionwill give an introduction to FGM and a brief overviewof Mindlin-Reissner plate theory Section 3 presents anoverview of the cell-based smoothed finite element methodwith discrete shear gap technique The efficiency of thepresent formulation numerical results and parametric stud-ies are presented in Section 4 followed by concludingremarks in the last section

2 Theoretical Background

21 Mindlin-Reissner Plate Theory The Mindlin-Reissnerplate theory also known as the first-order shear deformationtheory (FSDT) takes into account the shear deformationthrough the thickness Using the Mindlin formulation thedisplacements 119906 V and 119908 at a point (119909 119910 119911) in the plate(see Figure 1) from the medium surface are expressed asfunctions of the midplane displacements 119906

119900 V119900 and 119908

119900and

independent rotations 120579119909and 120579119910of the normal in 119910119911 and 119909119911

planes respectively as follows

119906 (119909 119910 119911 119905) = 119906119900(119909 119910 119905) + 119911120579

119909(119909 119910 119905)

V (119909 119910 119911 119905) = V119900(119909 119910 119905) + 119911120579

119910(119909 119910 119905)

119908 (119909 119910 119911 119905) = 119908119900(119909 119910 119905)

(1)

where 119905 is the time The strains in terms of midplanedeformation can be written as follows

120576 =

120576119901

0

+

119911120576119887

120576119904

(2)

The midplane strains 120576119901 the bending strain 120576

119887 and the shear

strain 120576119904in (2) are written as follows

120576119901=

119906119900119909

V119900119910

119906119900119910

+ V119900119909

120576119887=

120579119909119909

120579119910119910

120579119909119910

+ 120579119910119909

120576119904=

120579119909+ 119908119900119909

120579119910+ 119908119900119910

(3)

Mathematical Problems in Engineering 3

x

yz

a

b

h

(a)

a

b

y

120595

y998400

x x998400

(b)

Figure 1 (a) Coordinate system of a rectangular FGM plate and (b) coordinate system of a skew plate

where the subscript ldquocommardquo represents the partial derivativewith respect to the spatial coordinate succeeding itThemem-brane stress resultants N and the bending stress resultantsMcan be related to themembrane strains 120576

119901and bending strains

120576119887through the following constitutive relations

N =

119873119909119909

119873119910119910

119873119909119910

= A120576119901+ B120576119887minus Nth

M =

119872119909119909

119872119910119910

119872119909119910

= B120576119901+D119887120576119887minusMth

(4)

where the matrices A = 119860119894119895 B = 119861

119894119895 and D

119887= 119863119894119895 (119894 119895 =

1 2 6) are the extensional and bending-extensional couplingand bending stiffness coefficients and are defined as follows

119860119894119895 119861119894119895 119863119894119895 = int

ℎ2

minusℎ2

1198761198941198951 119911 119911

2

119889119911 (5)

Similarly the transverse shear force 119876 = 119876119909119911 119876119910119911 is

related to the transverse shear strains 120576119904through the following

equation

119876119894119895= 119864119894119895120576119904 (6)

whereE = 119864119894119895= int

ℎ2

minusℎ2

119876119894119895120592119894120592119895119889119911 (119894 119895 = 4 5) are the transverse

shear stiffness coefficients and 120592119894 120592119895is the transverse shear

coefficient for nonuniform shear strain distribution throughthe plate thickness The stiffness coefficients 119876

119894119895are defined

as

11987611=11987622

=

119864 (119911)

1 minus ]2 119876

12=

]119864 (119911)

1 minus ]2 119876

16= 11987626

= 0

11987644

= 11987655

= 11987666

=

119864 (119911)

2 (1 + ])

(7)

where the modulus of elasticity 119864(119911) and Poissonrsquos ratio ]are given by (20) The thermal stress resultant Nth and themoment resultantMth are

Nth=

119873th119909119909

119873th119910119910

119873th119909119910

= int

ℎ2

minusℎ2

119876119894119895120572 (119911 119879)

1

1

0

119879 (119911) 119889119911

Mth=

119872th119909119909

119872th119910119910

119872th119909119910

= int

ℎ2

minusℎ2

119876119894119895120572 (119911 119879)

1

1

0

119879 (119911) 119911 119889119911

(8)

where the thermal coefficient of expansion 120572(119911 119879) is givenby (21) 119879(119911) = 119879(119911) minus 119879

119900is the temperature rise from

the reference temperature and119879119900is the temperature at which

there are no thermal strains The strain energy function 119880 isgiven by

119880 (120575) =

1

2

int

Ω

120576119879

119901A120576119901+ 120576119879

119901B120576119887+ 120576119879

119887B120576119901

+120576119879

119887D120576119887+ 120576119879

119904E120576119904minus 120576119879

119887Nth

minus 120576119879

119887Mth

119889Ω

(9)

where 120575 = 119906 V 119908 120579119909 120579119910 is the vector of the degree of

freedom associated with the displacement field in a finiteelement discretization Following the procedure given in [40]the strain energy function 119880 given in (9) can be rewritten as

119880 (120575) =

1

2

120575119879K120575 (10)

where K is the linear stiffness matrix The kinetic energy ofthe plate is given by

119879 (120575) =

1

2

int

Ω

119901 (2

119900+ V2119900+ 2

119900) + 119868 (

1205792

119909+

1205792

119910) 119889Ω (11)

where = int

ℎ2

minusℎ2

120588(119911)119889119911 119868 = int

ℎ2

minusℎ2

1199112

120588(119911)119889119911 and 120588(119911) is themass density that varies through the thickness of the plateWhen the plate is subjected to a temperature field this in turnresults in in-plane stress resultants Nth The external work


due to the in-plane stress resultants developed in the plateunder a thermal load is given by

119881 (120575) = int

Ω

1

2

[119873th1199091199091199082

119909+ 119873

th1199101199101199082

119910+ 2119873

th119909119910119908119909119908119910]

+

ℎ2

24

[119873th119909119909

(1205792

119909119909+ 1205792

119910119909) + 119873

2

119910119910(1205792

119909119910+ 1205792

119910119910)

+ 2119873th119909119910

(120579119909119909

120579119909119910

+ 120579119910119909

120579119910119910

)] 119889Ω

(12)

Substituting (9)ndash(12) in Lagrangersquos equation of motion oneobtains the following finite element equations

Static bending

K120575 = F (13)

Free vibration

M 120575 + (K + K

119866) 120575 = 0 (14)

Buckling analysis

Mechanical buckling (prebuckling deforma-tions are assumed to be zero or negligible in theanalysis (including those coming from in-planeand out-of-plane coupling related to FGM andtemperature variation through the thickness ofthe plate))

(K + 120582119872K119866) 120575 = 0 (15)

Thermal buckling

(K + 120582119879K119866) 120575 = 0 (16)

where 120582119872

includes the critical value applied to in-planemechanical loading 120582

119879is the critical temperature difference

and K and K119866are the linear stiffness and geometric stiffness

matrices respectively The critical temperature difference iscomputed using a standard eigenvalue algorithm

22 Functionally Graded Material A rectangular plate madeof a mixture of ceramic and metal is considered with thecoordinates 119909 119910 along the in-plane directions and 119911 alongthe thickness direction (see Figure 1)Thematerial on the topsurface (119911 = ℎ2) of the plate is ceramic rich and is graded tometal at the bottom surface of the plate (119911 = minusℎ2) by a powerlawdistributionThe effective properties of the FGMplate canbe computed by using the rule of mixtures or by employingthe Mori-Tanaka homogenization scheme Let 119881

119894(119894 = 119888119898)

be the volume fraction of the phase materialThe subscripts 119888and 119898 refer to ceramic and metal phases respectively Thevolume fraction of ceramic and metal phases is related by119881119888+ 119881119898

= 1 and 119881119888is expressed as

119881119888(119911) = (

2119911 + ℎ

2ℎ

)

119899

(17)

where 119899 is the volume fraction exponent (119899 ge 0) also knownas the gradient index The variation of the composition ofceramic and metal is linear for 119899 = 1 the value of 119899 = 0

represents a fully ceramic plate and any other value of 119899 yieldsa composite material with a smooth transition from ceramicto metal

Mori-Tanaka Homogenization Method Based on the Mori-Tanaka homogenization method the effective Youngrsquos mod-ulus and Poissonrsquos ratio are computed from the effective bulkmodulus119870 and the effective shear modulus 119866 as [41]

119870eff minus 119870119898

119870119888minus 119870119898

=

119881119888

1 + 119881119898(3 (119870119888minus 119870119898) (3119870

119898+ 4119866119898))

119866eff minus 119866119898

119866119888minus 119866119898

=

119881119888

1 + 119881119898((119866119888minus 119866119898) (119866119898+ 1198911))

(18)

where

1198911=

119866119898(9119870119898+ 8119866119898)

6 (119870119898+ 2119866119898)

(19)

The effective Youngrsquos modulus 119864eff and Poissonrsquos ratio ]eff canbe computed from the following relations

119864eff =

9119870eff119866eff3119870eff + 119866eff

]eff =

3119870eff minus 2119866eff2 (3119870eff + 119866eff)

(20)

The effective mass density 120588eff is computed using the rule ofmixtures (120588eff = 120588

119888119881119888+120588119898119881119898) The effective heat conductivity

120581eff and the coefficient of thermal expansion 120572eff are given by

120581eff minus 120581119898

120581119888minus 120581119898

=

119881119888

1 + 119881119898((120581119888minus 120581119898) 3120581119898)

120572eff minus 120572119898

120572119888minus 120572119898

=

(1119870eff minus 1119870119898)

(1119870119888minus 1119870

119898)

(21)

Temperature-Dependent Material Property The materialproperties that are temperature dependent are written as [41]

119875 = 119875119900(119875minus1119879minus1

+ 1 + 1198751119879 + 11987521198792

+ 11987531198793

) (22)

where 119875119900 119875minus1 1198751 1198752 and 119875

3are the coefficients of temper-

ature 119879 and are unique to each constituent material phase

Temperature Distribution through the Thickness The temper-ature variation is assumed to occur in the thickness directiononly and the temperature field is considered to be constantin the 119909119910-plane In such a case the temperature distributionalong the thickness can be obtained by solving a steady stateheat transfer problem

minus

119889

119889119911

[120581 (119911)

119889119879

119889119911

] = 0 119879 =

119879119888

at 119911 =

ℎ

2

119879119898

at 119911 = minus

ℎ

2

(23)

The solution of (23) is obtained by means of a polynomialseries [42] as

119879 (119911) = 119879119898+ (119879119888minus 119879119898) 120578 (119911 ℎ) (24)


1

2

3

O

Δ1

Δ2

Δ3

Figure 2 A triangular element is divided into three subtriangles1 2 and

3are the subtriangles created by connecting the

central point 119874 with three field nodes

where

120578 (119911 ℎ) =

1

119862

[(

2119911 + ℎ

2ℎ

) minus

120581119888119898

(119899 + 1) 120581119898

(

2119911 + ℎ

2ℎ

)

119899+1

+

1205812

119888119898

(2119899 + 1) 1205812

119898

(

2119911 + ℎ

2ℎ

)

2119899+1

minus

1205813

119888119898

(3119899 + 1) 1205813

119898

(

2119911 + ℎ

2ℎ

)

3119899+1

+

1205814

119888119898

(4119899 + 1) 1205814

119898

(

2119911 + ℎ

2ℎ

)

4119899+1

minus

1205815

119888119898

(5119899 + 1) 1205815

119898

(

2119911 + ℎ

2ℎ

)

5119899+1

]

(25)

119862 = 1 minus

120581119888119898

(119899 + 1) 120581119898

+

1205812

119888119898

(2119899 + 1) 1205812

119898

minus

1205813

119888119898

(3119899 + 1) 1205813

119898

+

1205814

119888119898

(4119899 + 1) 1205814

119898

minus

1205815

119888119898

(5119899 + 1) 1205815

119898

(26)

where 120581119888119898

= 120581119888minus 120581119898

3 Cell-Based Smoothed Finite ElementMethod with Discrete Shear Gap Technique

In this study three-noded triangular element with fivedegrees of freedom (DOFs) 120575 = 119906 V 119908 120579

119909 120579119910 is employed

The displacement is approximated by

uℎ = sum

119868

119873119868120575119868 (27)

where 120575119868are the nodal DOFs and 119873

119868are the standard finite

element shape functions given by

119873 = [1 minus 120585 minus 120578 120578 120585] (28)

In this work the cell-based smoothed finite elementmethod (CSFEM) is combined with stabilized discrete sheargapmethod (DSG) for three-noded triangular element calledldquocell-based discrete shear gap method (CS-DSG3)rdquo [39] Thecell-based smoothing technique decreases the computationalcomplexity whilst DSG suppresses the shear locking phe-nomenon when the present formulation is applied to thinplates Interested readers are referred to the literature andreferences therein for the description of cell-based smoothingtechnique [24 26] and DSG method [43] In the CS-DSG3each triangular element is divided into three subtrianglesThedisplacement vector at the center node is assumed to be thesimple average of the three displacement vectors of the threefield nodes In each subtriangle the stabilized DSG3 is usedto compute the strains and also to avoid the transverse shearlocking Then the strain smoothing technique on the wholetriangular element is used to smooth the strains on the threesubtriangles Consider a typical triangular element Ω

119890as

shown in Figure 2This is first divided into three subtriangles1 2 and

3such that Ω

119890= ⋃3

119894=1119894 The coordinates of

the center point x119900= (119909119900 119910119900) are given by

(119909119900 119910119900) =

1

3

(119909119868 119910119868) (29)

The displacement vector of the center point is assumed to bea simple average of the nodal displacements as

120575119890119874

=

1

3

120575119890119868 (30)

The constant membrane strains the bending strains and theshear strains for subtriangle

1are given by

120576119901= [p11

p12

p13

]

120575119890119874

1205751198901

1205751198902

120576119887= [b11

b12

b13

]

120575119890119874

1205751198901

1205751198902

120576119904= [s11

s12

s13

]

120575119890119874

1205751198901

1205751198902

(31)

Upon substituting the expression for 120575119890119874

in (31) we obtain

1205761

119901= [

1

3

p11

+ p12

1

3

p11

+ p13

1

3

p11

]

times

1205751198901

1205751198902

1205751198903

= B1119901120575119890

1205761

119887= [

1

3

b11

+ b12

1

3

b11

+ b13

1

3

b11

]

times

1205751198901

1205751198902

1205751198903

= B1119887120575119890


1205761

119904= [

1

3

s11

+ s12

1

3

s11

+ s13

1

3

s11

]

times

1205751198901

1205751198902

1205751198903

= B1119904120575119890

(32)

where p119894 (119894 = 1 2 3) b

119894 (119894 = 1 2 3) and s

119894 (119894 = 1 2 3) are

given by

B119901=

1

2119860119890

[

[

[

[

119887 minus 119888 0 0 0 0

0 119889 minus 119886 0 0 0

119889 minus 119886 119887 minus 119888 0 0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

p1

119888 0 0 0 0

0 minus119889 0 0 0

minus119889 119888 0 0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

p2

minus119887 0 0 0 0

119886 0 0 0 0

119886 minus119887 0 0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

p3

]

]

]

]

B119887=

1

2119860119890

[

[

[

[

0 0 0 119887 minus 119888 0

0 0 0 0 119889 minus 119886

0 0 0 119889 minus 119886 119887 minus 119888⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

b1

0 0 0 119888 0

0 0 0 0 minus119889

0 0 0 minus119889 119888⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

b2

0 0 0 minus119887 0

0 0 0 0 119886

0 0 0 119886 minus119887⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

b3

]

]

]

]

B119904=

1

2119860119890

[

[

0 0 119887 minus 119888 119860119890

0

0 0 119889 minus 119886 0 119860119890⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

s1

0 0 119888 1198861198882 1198871198882

0 0 minus119889 minus1198861198892 minus1198871198892⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

s2

0 0 minus119887 minus1198871198892 minus1198871198882

0 0 119886 1198861198892 1198861198882⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

s3

]

]

(33)

where 119886 = 1199092minus 1199091 119887 = 119910

2minus 1199101 119888 = 119910

3minus 1199101 119889 = 119909

3minus 1199091

(see Figure 3) 119860119890is the area of the triangular element and

B119904is the altered shear strains [43] The strain-displacement

matrix for the other two triangles can be obtained by cyclicpermutation Now applying the cell-based strain smoothing[26] the constant membrane strains the bending strainsand the shear strains are respectively employed to create asmoothed membrane strain 120576

119901 smoothed bending strain 120576

119887

and smoothed shear strain 120576119904on the triangular elementΩ

119890as

follows

120576119901= int

Ω119890

120576119887Φ119890(x) 119889Ω =

3

sum

119894=1

120576119894

119901int

119894

Φ119890(x) 119889Ω

120576119887= int

Ω119890

120576119887Φ119890(x) 119889Ω =

3

sum

119894=1

120576119894

119887int

119894

Φ119890(x) 119889Ω

120576119904= int

Ω119890

120576119904Φ119890(x) 119889Ω =

3

sum

119894=1

120576119894

119904int

119894

Φ119890(x) 119889Ω

(34)

where Φ119890(x) is a given smoothing function that satisfies

Φ (x) =

1

119860119888

x isin Ω119888

0 x notin Ω119888

(35)

where119860119888is the area of the triangular elementThe smoothed

membrane strain the smoothed bending strain and thesmoothed shear strain are then given by

120576119901 120576119887 120576119904 =

sum3

119894=1119860119894

120576119894

119901 120576119894

119887 120576119894

119904

119860119890

(36)

The smoothed elemental stiffness matrix is given by

K = int

Ω119890

B119901AB119879119901+ B119901BB119879119887+ B119887BB119879119901+ B119887DB119879119887+ B119904EB119879119904119889Ω

= (B119901AB119879119901+ B119901BB119879119887+ B119887BB119879119901+ B119887DB119879119887+ B119904EB119879119904)119860119890

(37)

where B119901 B119887 and B

119904are the smoothed strain-displacement

matrix The mass matrix M is computed by following theconventional finite element procedure To further improvethe accuracy of the solution and to stabilize the shear forceoscillation the shear stiffness coefficients are multiplied bythe following factor

ShearFac =

ℎ3

ℎ2+ 120572ℎ2

119890

(38)

where 120572 is a positive constant and ℎ119890is the longest length of

the edge of an element

4 Numerical Examples

In this section we present the static bending response thelinear free vibration and the buckling analysis of FGMplates using cell-based smoothed finite element method withdiscrete shear gap techniqueThe effect of various parametersnamelymaterial gradient index 119899 skewness of the plate120595 theplate aspect ratio 119886119887 the plate thickness 119886ℎ and boundaryconditions on the global response is numerically studiedThetop surface of the plate is ceramic rich and the bottom surfaceof the plate is metal rich Here the modified shear correctionfactor obtained based on energy equivalence principle asoutlined in [44] is used The boundary conditions for simplysupported and clamped cases are as follows


b

b

c

d

120578

1

120585

2

3

Figure 3 Three-noded triangular element and local coordinates indiscrete shear gap method

simply supported boundary condition

119906119900= 119908119900= 120579119910= 0 on 119909 = 0 119886

V119900= 119908119900= 120579119909= 0 on 119910 = 0 119887

(39)

clamped boundary condition

119906119900= 119908119900= 120579119910= V119900= 120579119909= 0 on 119909 = 0 119886 119910 = 0 119887 (40)

Skew Boundary Transformation For skew plates the edges ofthe boundary elements may not be parallel to the global axes(119909 119910 119911) In order to specify the boundary conditions on skewedges it is necessary to use the edge displacements (1199061015840

119900 V1015840119900 1199081015840

119900)

and so forth in a local coordinate system (1199091015840

1199101015840

1199111015840

) (seeFigure 1) The element matrices corresponding to the skewedges are transformed from global axes to local axes onwhichthe boundary conditions can be conveniently specified Therelation between the global and the local degrees of freedomof a particular node is obtained by

120575 = L1198921205751015840

(41)

where 120575 and 1205751015840 are the generalized displacement vectorin the global and the local coordinate system respectivelyThe nodal transformation matrix for a node 119868 on the skewboundary is given by

L119892=

[

[

[

[

[

[

cos120595 sin120595 0 0 0

minus sin120595 cos120595 0 0 0

0 0 1 0 0

0 0 0 cos120595 sin120595

0 0 0 minus sin120595 cos120595

]

]

]

]

]

]

(42)

where 120595 defines the skewness of the plate

41 Static Bending Let us consider a AlZrO2FGM square

plate with length-to-thickness 119886ℎ = 5 subjected to a uniformloadwith fully simply supported (SSSS) boundary conditions

Table 1 The normalized center deflection 119908119888= 100119908

119888(119864119888ℎ3

12(1 minus

]2)1199011198864) for a simply supported AlZrO2-1 FGM square plate with119886ℎ = 5 subjected to a uniformly distributed load 119901

Method Gradient index 1198990 1 2

4 times 4 01443 02356 026448 times 8 01648 02703 0302916 times 16 01701 02795 0313132 times 32 01714 02819 0315840 times 40 01716 02822 03161NS-DSG3 [47] 01721 02716 03107ES-DSG3 [47] 01700 02680 03066MLPG [45] 01671 02905 03280119896119901-Ritz [46] 01722 02811 03221MITC4 [47] 01715 02704 03093IGA-Quadratic [22] 01717 02719 03115

The Youngrsquos modulus for ZrO2

is 119864119888

= 151 GPa and foraluminum is 119864

119898= 70GPa Poissonrsquos ratio is chosen as

constant ] = 03 Table 1 compares the results from thepresent formulation with other approaches available in theliterature [22 45ndash47] and a very good agreement can beobserved Next we illustrate the performance of the presentformulation for thin plate problems A simply supportedsquare plate subjected to uniform load is considered whilethe length-to-thickness (119886ℎ) varies from 5 to 10

4Three indi-vidual approaches are employed discrete shear gap methodreferred to as DSG3 the cell-based smoothed finite elementmethod with discrete shear gap technique (CSDSG3) andthe stabilized CSDSG3 (where the shear stiffness coefficientsare multiplied by the stabilization factor) The normalizedcenter deflection 119908

119888= 100119908

119888(119864119898ℎ3

12(1 minus ]2)1199011198864) isshown in Figure 4 It is observed that the DSG3 results aresubjected to shear locking when the plate becomes thin(119886ℎ gt 100) However the present formulation CSDSG3with stabilization is less sensitive to shear locking

42 Free Flexural Vibrations In this section the free flexuralvibration characteristics of FGM plates with and withoutcentrally located cutout in thermal environment are studiednumerically Figure 5 shows the geometry of the plate witha centrally located circular cutout In all cases we presentthe nondimensionalized free flexural frequency defined asfollows unless otherwise stated

120596 = 1205961198862

radic

120588119888ℎ

119863119888

(43)

where 120596 is the natural frequency and 120588119888 119863119888= 119864119888ℎ3

12(1 minus

]2) are the mass density and the flexural rigidity of theceramic phase The FGM plate considered here is made upof silicon nitride (Si

3N4) and stainless steel (SUS304) The

material is considered to be temperature dependent and thetemperature coefficients corresponding to Si

3N4SUS304 are

listed in Table 2 [13 41]Themass density (120588) and the thermalconductivity (120581) are 120588

119888= 2370 kgm3and 120581

119888= 919WmK


Table 2 Temperature-dependent coefficient for material Si3N4SUS304 [13 41]

Material Property 119875119900

119875minus1

1198751

1198752

1198753

Si3N4119864 (Pa) 34843119890

9 00 minus3070119890minus4

2160119890minus7

minus8946119890minus11

120572 (1K) 58723119890minus6 00 9095119890

minus4 00 00

SUS304 119864 (Pa) 201041198909 00 3079119890

minus4

minus6534119890minus7 00

120572 (1K) 12330119890minus6 00 8086119890

minus4 00 00

002

004

006

008

01

012

014

016

018

Nor

mal

ized

cent

ral d

eflec

tion

DSG3CSDSG3CSDSG3 with stabilization

ah

100 101 102 103 104

Figure 4 The normalized center deflection as a function ofnormalized plate thickness for a simply supported square FGMplatesubjected to a uniform load

for Si3N4and 120588

119898= 8166 kgm3and 120581

119898= 1204WmK for

SUS304 Poissonrsquos ratio ] is assumed to be constant and takenas 028 for the current study [41] Before proceeding with adetailed study on the effect of gradient index on the naturalfrequencies the formulation developed herein is validatedagainst available analyticalnumerical solutions pertaining tothe linear frequencies of a FGMplate in thermal environmentand a FGM plate with a centrally located circular cutoutThe computed frequencies (a) for a square simply supportedFGM plate in thermal environment with 119886ℎ = 10 are givenin Table 3 and (b) the mesh convergence and comparison oflinear frequencies for a square plate with circular cutout aregiven in Tables 4 and 5 It can be seen that the numericalresults from the present formulation are found to be in verygood agreement with the existing solutions For the uniformtemperature case the material properties are evaluated at119879119888

= 119879119898

= 300K The temperature is assumed to varyonly in the thickness direction and is determined by (24)The temperature for the ceramic surface is varied whilst aconstant value on the metallic surface is maintained (119879

119898=

300K) to subject a thermal gradient The geometric stiffnessmatrix is computed from the in-plane stress resultants due tothe applied thermal gradient The geometric stiffness matrixis then added to the stiffness matrix and the eigenvalue

a

b

r

Figure 5 Platewith a centrally located circular cutout 119903 is the radiusof the circular cutout

problem is solved The effect of the material gradient index isalso shown in Tables 3 and 5 and the influence of a centrallylocated cutout is shown in Tables 4 and 5 The combinedeffect of increasing the temperature and the gradient indexis to lower the fundamental frequency and this is due to theincrease in the metallic volume fraction Figure 6 shows theinfluence of the cutout size on the frequency for a plate inthermal environment (119879 = 100K) The frequency increaseswith increasing cutout size This can be attributed to thedecrease in stiffness due to the presence of the cutout Also itcan be seen that with increasing gradient index the frequencydecreases In this case the decrease in the frequency isdue to the increase in the metallic volume fraction It isobserved that the combined effect of increasing the gradientindex and the cutout size is to lower the fundamentalfrequency Increasing the thermal gradient further decreasesthe fundamental frequency

43 Buckling Analysis In this section we present themechanical and thermal buckling behaviour of functionallygraded skew plates

Mechanical Buckling The FGM plate considered here con-sists of aluminum (Al) and zirconium dioxide (ZrO

2) The

material is considered to be temperature independent TheYoungrsquos modulus (119864) for ZrO

2is 119864119888

= 151 GPa and for Alis 119864119898

= 70GPa For mechanical buckling we consider both


Table 3 The first normalized frequency parameter 120596 for a fully simply supported Si3N4SUS304 FGM square plate with 119886ℎ = 10 in thermalenvironment

119879119888 119879119898

Gradient index 1198990 1 5 10

300 300 Present 183570 110690 90260 85880[48] 183731 110288 90128 85870

400 300 Present 179778 107979 88626 83182[48] 179620 107860 87530 83090

600 300 Present 171205 101679 81253 76516[48] 171050 101550 81150 76420

Table 4 Convergence of fundamental frequency (Ω =

[1205962

120588119888ℎ1198864

119863119888(1 minus ]2)]

14

) with mesh size for an isotropic plate witha central cutout

Number of nodes Mode 1 Mode 2333 61025 86297480 60805 85595719 60663 851921271 60560 84852[49] 61725 86443[50] 62110 87310

0 005 01 015 02 025 03 035 045

10

15

20

25

30

35

40

Non

dim

ensio

naliz

ed m

ode 1

freq

uenc

y

ra

n = 0n = 5

Figure 6 Effect of the cutout size on the fundamental frequency(Ω) for a square simply supported FGM plate with a central circularcutout in thermal environment 119879 = 100K (119879

119888= 400K 119879

119898=

300K) for different gradient index 119899

uni- and biaxial mechanical loads on the FGM plates In allcases we present the critical buckling parameters as followsunless otherwise specified

120582cru =

1198730

119909119909cr1198872

1205872119863119888

120582crb =

1198730

119910119910cr1198872

1205872119863119888

(44)

05 1 15 2 251

2

3

4

5

6

7

ab

n = 0

n = 5

Criti

cal b

uckl

ing

load

120582cr

u

Figure 7 Effect of plate aspect ratio 119886119887 and gradient index onthe critical buckling load for a simply supported FGM plate underuniaxial compression with 119886ℎ = 10

where 120582cru and 120582crb are the critical buckling parameters foruni- and biaxial load respectively 119863

119888= 119864119888ℎ3

(12(1 minus ]2))The critical buckling loads evaluated by varying the skewangle of the plate and volume fraction index and consideringmechanical loads (uni- and biaxial compressive loads) areshown in Table 6 for 119886ℎ = 100 The efficacy of the presentformulation is demonstrated by comparing our results withthose in [14] It can be seen that increasing the gradient indexdecreases the critical buckling load A very good agreementin the results can be observed It is also observed that thedecrease in the critical value is significant for the materialgradient index 119899 le 2 and that further increase in 119899 yieldsless reduction in the critical value irrespective of the skewangle The effect of the plate aspect ratio and the gradientindex on the critical buckling load is shown in Figure 7 fora simply supported FGM plate under uniaxial mechanicalload It is observed that the combined effect of increasingthe gradient index and the plate aspect ratio is to lower thecritical buckling load Table 7 presents the critical bucklingparameter for a simply supported FGM with a centrallylocated circular cutout with 119903119886 = 02 It can be seen that the


Table 5 Comparison of fundamental frequency for a simply supported FGM plate with 119886ℎ = 5 and 119903119886 = 02

119879119888

Gradient index 1198990 1 2 5 10

300 [49] 176855 106681 96040 87113 82850Present 177122 106845 96188 87246 82976

400 [37] 174690 105174 94618 85738 81484Present 175488 105775 95197 86309 82059

Table 6 Critical buckling parameters for a thin simply supported FGM skew plate with 119886ℎ = 100 and 119886119887 = 1

Skew angle 120582cr


[14] Present [43] Present

0∘ 120582cru 40010 40034 17956 18052 12624 10846120582crb 20002 20017 08980 09028 06312 05423

15∘ 120582cru 43946 44007 19716 19799 13859 11915120582crb 21154 21187 09517 09561 06683 05741

30∘ 120582cru 58966 59317 26496 26496 18586 16020120582crb 25365 25491 11519 11520 08047 06909

Table 7 Comparison of critical buckling load 120582cru =

119873119900

119909119909cr 1198872

1205872

119863119898

for a simply supported FGM plate with 119886ℎ =

100 and 119903119886 = 02 The effective material properties are computedby rule of mixtures In order to be consistent with the literature theproperties of the metallic phase are used for normalization

Gradient index 119899 [51] Present difference0 52611 52831 minus04202 46564 46919 minus0761 36761 3663 0362 33672 33961 minus0865 31238 31073 05310 29366 28947 143

present formulation yields comparable results The effect ofincreasing the gradient index is to lower the critical bucklingload This can be attributed to the stiffness degradation dueto increase in the metallic volume fraction Figure 8 showsthe influence of a centrally located circular cutout and thegradient index on the critical buckling load under two differ-ent boundary conditions namely all edges simply supportedand all edges clamped In this case the plate is subjected toa uniaxial compressive load It can be seen that increasingthe gradient index decreases the critical buckling load dueto increasing metallic volume fraction whilst increasing thecutout radius decreases the critical buckling load in thecase of simply supported boundary conditions This can beattributed to the stiffness degradation due to the presence of acutout and the simply supported boundary condition In caseof the clamped boundary condition the critical buckling loadfirst decreases with increasing cutout radius due to stiffnessdegradation Upon further increase the critical buckling loadincreases This is because the clamped boundary conditionadds stiffness to the system which overcomes the stiffnessreduction due to the presence of a cutout

0 005 01 015 02 025 03 035 040

2

4

6

8

10

12

14

Criti

cal b

uckl

ing

load

120582un

i

ra

SSSS (n = 0)SSSS (n = 5)

CCCC (n = 0)CCCC (n = 5)

Figure 8 Variation of the critical buckling load 120582cru =

119873119900

119909119909cr1198872

1205872

119863119888 with cutout dimensions for a square FGM plate with

119886ℎ = 10 subjected to uniaxial compressive loading for differentgradient index 119899 and various boundary conditions

Thermal BucklingThe thermal buckling behaviour of simplysupported functionally graded skew plate is studied nextThetop surface is ceramic rich and the bottom surface is metalrich The FGM plate considered here consists of aluminumand alumina Youngrsquosmodulus the thermal conductivity andthe coefficient of thermal expansion for alumina are 119864

119888=

380GPa 120581119888=104WmK and 120572

119888= 74 times 10

minus6 1∘C and foraluminum 119864

119898= 70GPa 120581

119898= 204WmK and 120572

119898= 23 times

10minus6 1∘C respectively Poissonrsquos ratio is chosen as constant

] = 03 The temperature rise of 119879119898

= 5∘C in the metal-rich surface of the plate is assumed in the present study


Table 8 Convergence of the critical buckling temperature for a simply supported FGM skew plate with 119886ℎ = 10 and 119886119887 = 1 Nonlineartemperature rise through the thickness of the plate is assumed

Mesh Gradient index 1198990 1 5 10

8 times 8 338340 205461 153924 14963616 times 16 328690 199507 149525 14539932 times 32 326391 198096 148476 14438640 times 40 326117 197930 148351 144260[9] 325747 197701 148183 144102

05 1 15 2 25500

1000

1500

2000

2500

3000

3500

LinearNonlinear

ab

Criti

cal b

uckl

ing

tem

pera

ture

ΔT

cr

Figure 9 Critical buckling temperature as a function of plateaspect ratio 119886119887 with linear and nonlinear temperature distributionthrough the thickness with gradient index 119899 = 5

In addition to nonlinear temperature distribution across theplate thickness the linear case is also considered in thepresent analysis by truncating the higher order terms in (25)The plate is of uniform thickness and simply supported onall four edges Table 8 shows the convergence of the criticalbuckling temperature with mesh size for different gradientindex 119899 It can be seen that the results from the presentformulation are in very good agreement with the availablesolution The influence of the plate aspect ratio 119886119887 and theskew angle120595 on the critical buckling temperature for a simplysupported square FGM plate are shown in Figures 9 and 10It is seen that increasing the plate aspect ratio decreases thecritical buckling temperature for both linear and nonlineartemperature distribution through the thickness The criticalbuckling temperature increases with increase in the skewangle The influence of the gradient index 119899 is also shownin Figure 10 It is seen that with increasing gradient index119899 the critical buckling temperature decreases This is due tothe increase in the metallic volume fraction that degradesthe overall stiffness of the structure Figure 11 shows theinfluence of the cutout radius and the material gradientindex on the critical buckling temperature Both linear and

0 5 10 15 20 25 301000

1500

2000

2500

3000

3500

4000

4500

Skew angle 120595

n = 0 (linear temperature)n = 0 (nonlinear temperature)n = 5 (linear temperature)n = 5 (nonlinear temperature)

Criti

cal b

uckl

ing

tem

pera

ture

ΔT

cr

Figure 10 Critical buckling temperature as a function of skew angle120595 for a simply supported square FGM plate with 119886ℎ = 10 Bothlinear and nonlinear temperature distribution through the thicknessare assumed

nonlinear temperature distribution through the thicknessare assumed Again it is seen that the combined effect ofincreasing the gradient index 119899 and the cutout radius 119903119886

is to lower the buckling temperature For gradient index119899 = 0 there is no difference between the linear and thenonlinear temperature distribution through the thickness asthematerial is homogeneous through the thickness while for119899 gt 0 the material is heterogeneous through the thicknesswith different thermal property

5 Conclusion

In this paper we applied the cell-based smoothed finiteelement method with discrete shear gap technique to studythe static and the dynamic response of functionally gradedmaterials The first-order shear deformation theory was usedto describe the plate kinematics The efficiency and accuracyof the present approach is demonstrated with few numericalexamples This improved finite element technique showsinsensitivity to shear locking and produces excellent results


0 005 01 015 02 025 03 035 04500

1000

1500

2000

2500

3000

3500

Criti

cal b

uckl

ing

tem

pera

ture

ra

n = 0

n = 5 (linear)n = 5 (nonlinear)

Figure 11 Influence of cutout size on the critical buckling tem-perature for a square simply supported FGM plate with a centrallylocated circular cutout with 119886ℎ = 10 for various gradient indices119899 Linear and nonlinear temperature distribution through thethickness are assumed

in static bending free vibration and buckling of functionallygraded plates

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

S Natarajan would like to acknowledge the financial supportof the School of Civil and Environmental EngineeringUniversity of New South Wales for his research fellowshipsince September 2012

References

[1] M Koizumi ldquoThe concept of FGMrdquo Ceramic Transactions vol34 pp 3ndash10 1993

[2] A M Zenkour ldquoGeneralized shear deformation theory forbending analysis of functionally graded platesrdquo Applied Mathe-matical Modelling vol 30 no 1 pp 67ndash84 2006

[3] A M Zenkour ldquoBenchmark trigonometric and 3-D elasticitysolutions for an exponentially graded thick rectangular platerdquoArchive of Applied Mechanics vol 77 no 4 pp 197ndash214 2007

[4] J N Reddy ldquoAnalysis of functionally graded platesrdquo Interna-tional Journal for Numerical Methods in Engineering vol 47 no1ndash3 pp 663ndash684 2000

[5] M K Singha T Prakash and M Ganapathi ldquoFinite elementanalysis of functionally graded plates under transverse loadrdquoFinite Elements in Analysis and Design vol 47 no 4 pp 453ndash460 2011

[6] S Natarajan PM Baiz S Bordas T Rabczuk and P KerfridenldquoNatural frequencies of cracked functionally graded materialplates by the extended finite element methodrdquo CompositeStructures vol 93 no 11 pp 3082ndash3092 2011

[7] G N Praveen and J N Reddy ldquoNonlinear transient ther-moelastic analysis of functionally graded ceramic-metal platesrdquoInternational Journal of Solids and Structures vol 35 no 33 pp4457ndash4476 1998

[8] K YDai G R Liu XHan andKM Lim ldquoThermomechanicalanalysis of functionally graded material (FGM) plates usingelement-free Galerkin methodrdquo Computers and Structures vol83 no 17-18 pp 1487ndash1502 2005

[9] MGanapathi and T Prakash ldquoThermal buckling of simply sup-ported functionally graded skew platesrdquo Composite Structuresvol 74 no 2 pp 247ndash250 2006

[10] M Janghorban and A Zare ldquoThermal effect on free vibrationanalysis of functionally graded arbitrary straight-sided plateswith different cutoutsrdquo Latin American Journal of Solids andStructures vol 8 no 3 pp 245ndash257 2011

[11] A M Zenkour and D S Mashat ldquoThermal buckling analysis ofceramic-metal functionally graded platesrdquo Natural Science vol2 pp 968ndash978 2010

[12] X Zhao Y Y Lee and K M Liew ldquoFree vibration analysisof functionally graded plates using the element-free kp-Ritzmethodrdquo Journal of Sound and Vibration vol 319 no 3ndash5 pp918ndash939 2009

[13] J N Reddy and C D Chin ldquoThermomechanical analysisoffunctionally graded cylinders and platesrdquo Journal of ThermalStresses vol 21 no 6 pp 593ndash626 1998

[14] M Ganapathi T Prakash and N Sundararajan ldquoInfluence offunctionally graded material on buckling of skew plates undermechanical loadsrdquo Journal of Engineering Mechanics vol 132no 8 pp 902ndash905 2006

[15] B A Samsam Shariat and M R Eslami ldquoThermal buckling ofimperfect functionally graded platesrdquo International Journal ofSolids and Structures vol 43 no 14-15 pp 4082ndash4096 2006

[16] D Saji B Varughese and S C Pradhan ldquoFinite element analy-sis for thermal buckling behaviour in functionally graded plateswith cutoutsrdquo in Proceedings of the International conference onAerospace Science andTechnology (INCAST rsquo08) vol 113 pp 26ndash28 Bangalore India Jun

[17] E Carrera S Brischetto and A Robaldo ldquoVariable kinematicmodel for the analysis of functionally graded material platesrdquoAIAA Journal vol 46 no 1 pp 194ndash203 2008

[18] E Carrera S Brischetto M Cinefra and M Soave ldquoEffects ofthickness stretching in functionally graded plates and shellsrdquoComposites N vol 42 no 2 pp 123ndash133 2011

[19] M Cinefra E Carrera L Della Croce and C Chinosi ldquoRefinedshell elements for the analysis of functionally graded structuresrdquoComposite Structures vol 94 pp 415ndash422 2012

[20] A J M Ferreira R C Batra C M C Roque L F Qian and RM N Jorge ldquoNatural frequencies of functionally graded platesby a meshless methodrdquo Composite Structures vol 75 no 1ndash4pp 593ndash600 2006

[21] L C Qian R C Batra and L M Chen ldquoStatic and dynamicdeformations of thick functionally graded elastic plates by usinghigher order shear and normal deformable plate theory andmeshless local Petrov Galerkin methodrdquo Composites B vol 35pp 685ndash697 2004

[22] N Valizadeh S Natarajan O A Gonzalez-Estrada T RabczukT Q Bui and S Bordas ldquoNURBS-based finite element analysis


of functionally graded plates elastic bending vibraton buck-ling and flutterrdquoComposite Structures vol 99 pp 209ndash326 2013

[23] V T Loc H T Chien and H Nguyen-Xuan ldquoAn isogeometricfinite element formulation for thermal buckling analysis offunctionally graded platesrdquo Finite Elements in Analysis andDesign vol 73 pp 65ndash76 2013

[24] G R Liu K Y Dai and T T Nguyen ldquoA smoothed finiteelement method for mechanics problemsrdquo ComputationalMechanics vol 39 no 6 pp 859ndash877 2007

[25] H Nguyen-Xuan S Bordas and H Nguyen-Dang ldquoSmoothfinite element methods convergence accuracy and propertiesrdquoInternational Journal for Numerical Methods in Engineering vol74 no 2 pp 175ndash208 2008

[26] S P A Bordas and S Natarajan ldquoOn the approximation in thesmoothed finite elementmethod (SFEM)rdquo International Journalfor Numerical Methods in Engineering vol 81 no 5 pp 660ndash670 2010

[27] G R Liu T Nguyen-Thoi H Nguyen-Xuan and K Y LamldquoA node-based smoothed finite element method (NS-FEM) forupper bound solutions to solid mechanics problemsrdquo Comput-ers and Structures vol 87 no 1-2 pp 14ndash26 2009

[28] G R Liu T Nguyen-Thoi and K Y Lam ldquoAn edge-basedsmoothed finite element method (ES-FEM) for static freeand forced vibration analyses of solidsrdquo Journal of Sound andVibration vol 320 no 4-5 pp 1100ndash1130 2009

[29] T Nguyen-Thoi G R Liu K Y Lam and G Y Zhang ldquoAface-based smoothed finite element method (FS-FEM) for 3Dlinear and geometrically non-linear solid mechanics problemsusing 4-node tetrahedral elementsrdquo International Journal forNumerical Methods in Engineering vol 78 no 3 pp 324ndash3532009

[30] G R Liu T Nguyen-Thoi and K Y Lam ldquoA novel alphafinite element method (120572FEM) for exact solution to mechanicsproblems using triangular and tetrahedral elementsrdquo ComputerMethods in Applied Mechanics and Engineering vol 197 no 45ndash48 pp 3883ndash3897 2008

[31] F Cazes and G Meschke ldquoAn edge based imbricate finiteelement method (EI-FEM) with full and reduced integrationrdquoComputer and Structures vol 106-107 pp 154ndash175 2012

[32] H Nguyen-Xuan T Rabczuk S Bordas and J F Debongnie ldquoAsmoothed finite element method for plate analysisrdquo ComputerMethods in Applied Mechanics and Engineering vol 197 no 13ndash16 pp 1184ndash1203 2008

[33] N Nguyen-Thanh T Rabczuk H Nguyen-Xuan and S P ABordas ldquoA smoothed finite element method for shell analysisrdquoComputer Methods in Applied Mechanics and Engineering vol198 no 2 pp 165ndash177 2008

[34] S C Wu G R Liu X Y Cui T T Nguyen and G Y ZhangldquoAn edge-based smoothed point interpolation method (ES-PIM) for heat transfer analysis of rapid manufacturing systemrdquoInternational Journal of Heat andMass Transfer vol 53 no 9-10pp 1938ndash1950 2010

[35] H Nguyen-Xuan G R Liu S Bordas S Natarajan and TRabczuk ldquoAn adaptive singular ES-FEM for mechanics prob-lemswith singular field of arbitrary orderrdquoComputerMethods inApplied Mechanics and Engineering vol 253 pp 252ndash273 2013

[36] Z C He A G Cheng G Y Zhang Z H Zhong and G RLiu ldquoDispersion error reduction for acoustic problems usingthe edge-based smoothed finite element method (ES-FEM)rdquoInternational Journal for Numerical Methods in Engineering vol86 no 11 pp 1322ndash1338 2011

[37] S P A Bordas S Natarajan P Kerfriden et al ldquoOn theperformance of strain smoothing for quadratic and enrichedfinite element approximations (XFEMGFEMPUFEM)rdquo Inter-national Journal for Numerical Methods in Engineering vol 86no 4-5 pp 637ndash666 2011

[38] D K Jha T Kant and R K Singh ldquoA critical review of recentresearch on functionally graded platesrdquo Composite Structuresvol 96 pp 833ndash849 2013

[39] T Nguyen-Thoi P Phung-Van H Nguyen-Xuan and C Thai-Hoang ldquoA cell-based smoothed discrete shear gap methodusing triangular elements for static and free vibration analysesof Reissner-Mindlin platesrdquo International Journal for NumericalMethods in Engineering vol 91 no 7 pp 705ndash741 2012

[40] S Rajesekaran and D W Murray ldquoIncremental finite elementmatricesrdquoASCE Journal of Structural Divison vol 99 no 12 pp2423ndash2438 1973

[41] N Sundararajan T Prakash andMGanapathi ldquoNonlinear freeflexural vibrations of functionally graded rectangular and skewplates under thermal environmentsrdquo Finite Elements in Analysisand Design vol 42 no 2 pp 152ndash168 2005

[42] W Lanhe ldquoThermal buckling of a simply supportedmoderatelythick rectangular FGM platerdquo Composite Structures vol 64 no2 pp 211ndash218 2004

[43] K-U Bletzinger M Bischoff and E Ramm ldquoUnified approachfor shear-locking-free triangular and rectangular shell finiteelementsrdquo Computers and Structures vol 75 no 3 pp 321ndash3342000


[45] D F Gilhooley R C Batra J R Xiao M A McCarthy and JW Gillespie Jr ldquoAnalysis of thick functionally graded plates byusing higher-order shear and normal deformable plate theoryand MLPG method with radial basis functionsrdquo CompositeStructures vol 80 no 4 pp 539ndash552 2007

[46] Y Y Lee X Zhao and K M Liew ldquoThermoelastic analysisof functionally graded plates using the element-free kp-Ritzmethodrdquo Smart Materials and Structures vol 18 no 3 ArticleID 035007 2009

[47] H Nguyen-Xuan L V Tran C H Thai and T Nguyen-ThoildquoAnalysis of functionally graded plates by an efficient finiteelement method with node-based strain smoothingrdquo Thin-Walled Structures vol 54 pp 1ndash18 2012

[48] S Natarajan P M Baiz M Ganapathi P Kerfriden and SBordas ldquoLinear free flexural vibration of cracked functionallygraded plates in thermal environmentrdquo Computers and Struc-tures vol 89 no 15-16 pp 1535ndash1546 2011

[49] A A Rahimabadi S Natarajan and S Bordas ldquoVibration offunctionally graded material plates with cutouts and cracks inthermal environmentrdquo Key Engineering Materials vol 560 pp157ndash180 2013

[50] M Huang and T Sakiyama ldquoFree vibration analysis of rectan-gular plates with variously-shaped holesrdquo Journal of Sound andVibration vol 226 no 4 pp 769ndash786 1999

[51] X Zhao Y Y Lee and K M Liew ldquoMechanical and thermalbuckling analysis of functionally graded platesrdquo CompositeStructures vol 90 no 2 pp 161ndash171 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


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Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


equation Their study concluded that the structures madeup of FGM with ceramic rich side exposed to elevatedtemperatures are susceptible to buckling due to the through-thickness temperature variation The buckling of skewedFGM plates under mechanical and thermal loads was studied[9 14] by employing the first-order shear deformation theoryand by using the shear flexible quadrilateral element Effortshave also been made to study the mechanical behaviour ofFGM plates with geometrical imperfection [15] Saji et al[16] have studied thermal buckling of FGM plates withmaterial properties dependent on both the composition andtemperature They found that the critical buckling temper-atures decreased when material properties are consideredto be a function of temperature as compared to the resultsobtained where the material properties are assumed to beindependent of temperature Ganapathi and Prakash [9] havestudied the buckling of FGM skewed plate under thermalloading Efforts have also been made to study the mechanicalbehaviour of FGM plates with geometrical imperfection [15]More recently refinedmodels have been adopted to study thecharacteristics of FGM structures [17ndash19]

Existing approaches in the literature to study plate andshell structures made up of FGMs use finite element method(FEM) based on Lagrange basis functions [14] and meshfreemethods [20 21] and recently Valizadeh et al [22] usednonuniform rational B-splines based FEM to study thestatic and dynamic characteristics of FGM plates in thermalenvironment Loc et al [23] employed isogeometric finiteelement method to study thermal buckling of functionallygraded plates Even with these different approaches theplate elements suffer from shear locking phenomenon anddifferent techniques were proposed to alleviate the shearlocking phenomenon Another set of methods have emergedto address the shear locking in the FEM By incorporating thestrain smoothing technique into the finite element method(FEM) Liu et al [24] have formulated a series of smoothedfinite element methods (SFEM) named as cell-based SFEM(CS-FEM) [25 26] node-based SFEM [27] edge-basedSFEM [28] face-based SFEM [29] and 120572-FEM [30] Andrecently edge based imbricate finite element method (EI-FEM) was proposed in [31] that shares common features withthe ES-FEM As the SFEM can be recast within a Hellinger-Reissner variational principle suitable choices of the assumedstraingradient space provide stable solutions Depending onthe number and geometry of the subcells used a spectrumof methods exhibiting a spectrum of properties is obtainedInterested readers are referred to the literature [24 25] andreferences therein Nguyen-Xuan et al [32] employed CS-FEM forMindlin-Reissner platesThe curvature at each pointis obtained by a nonlocal approximation via a smoothingfunction From the numerical studies presented it was con-cluded that the CS-FEM technique is robust computationallyinexpensive free of locking andmost importantly insensitiveto mesh distortions The SFEM was extended to variousproblems such as shells [33] heat transfer [34] fracturemechanics [35] and structural acoustics [36] among othersIn [37] CS-FEM has been combined with the extended FEMto address problems involving discontinuities The above listis no way comprehensive and interested readers are referred

to the literature and references therein and to a recent reviewpaper by Jha et al [38] on FGM plates

Approach In this paper we study the static and the dynamiccharacteristics of FGM plates by using a cell-based smoothedfinite element method with discrete shear gap technique [39]Three-noded triangular element is employed in this studyThe effect of different parameters namely the material gra-dient index the plate aspect ratio the plate slenderness ratioand the boundary condition on the global response of FGMplates is numerically studied The effect of centrally locatedcircular cutout is also studied The present work focuses onthe computational aspects of the governing equations andhence the attention has been restricted to Mindlin-Reissnerplate theory It is noted that the extension to higher ordertheories is possible

Outline The paper is organized as follows the next sectionwill give an introduction to FGM and a brief overviewof Mindlin-Reissner plate theory Section 3 presents anoverview of the cell-based smoothed finite element methodwith discrete shear gap technique The efficiency of thepresent formulation numerical results and parametric stud-ies are presented in Section 4 followed by concludingremarks in the last section

2 Theoretical Background

21 Mindlin-Reissner Plate Theory The Mindlin-Reissnerplate theory also known as the first-order shear deformationtheory (FSDT) takes into account the shear deformationthrough the thickness Using the Mindlin formulation thedisplacements 119906 V and 119908 at a point (119909 119910 119911) in the plate(see Figure 1) from the medium surface are expressed asfunctions of the midplane displacements 119906

119900 V119900 and 119908

119900and

independent rotations 120579119909and 120579119910of the normal in 119910119911 and 119909119911

planes respectively as follows

119906 (119909 119910 119911 119905) = 119906119900(119909 119910 119905) + 119911120579

119909(119909 119910 119905)

V (119909 119910 119911 119905) = V119900(119909 119910 119905) + 119911120579

119910(119909 119910 119905)

119908 (119909 119910 119911 119905) = 119908119900(119909 119910 119905)

(1)

where 119905 is the time The strains in terms of midplanedeformation can be written as follows

120576 =

120576119901

0

+

119911120576119887

120576119904

(2)

The midplane strains 120576119901 the bending strain 120576

119887 and the shear

strain 120576119904in (2) are written as follows

120576119901=

119906119900119909

V119900119910

119906119900119910

+ V119900119909

120576119887=

120579119909119909

120579119910119910

120579119909119910

+ 120579119910119909

120576119904=

120579119909+ 119908119900119909

120579119910+ 119908119900119910

(3)


x

yz

a

b

h

(a)

a

b

y

120595

y998400

x x998400

(b)





N =

119873119909119909

119873119910119910

119873119909119910

= A120576119901+ B120576119887minus Nth

M =

119872119909119909

119872119910119910

119872119909119910

= B120576119901+D119887120576119887minusMth

(4)


119894119895 and D

119887= 119863119894119895 (119894 119895 =


119860119894119895 119861119894119895 119863119894119895 = int

ℎ2

minusℎ2

1198761198941198951 119911 119911

2

119889119911 (5)



equation

119876119894119895= 119864119894119895120576119904 (6)

whereE = 119864119894119895= int

ℎ2

minusℎ2





as

11987611=11987622

=

119864 (119911)

1 minus ]2 119876

12=

]119864 (119911)

1 minus ]2 119876

16= 11987626

= 0

11987644

= 11987655

= 11987666

=

119864 (119911)

2 (1 + ])

(7)


Nth=

119873th119909119909

119873th119910119910

119873th119909119910

= int

ℎ2

minusℎ2

119876119894119895120572 (119911 119879)

1

1

0

119879 (119911) 119889119911

Mth=

119872th119909119909

119872th119910119910

119872th119909119910

= int

ℎ2

minusℎ2

119876119894119895120572 (119911 119879)

1

1

0

119879 (119911) 119911 119889119911

(8)





119880 (120575) =

1

2

int

Ω

120576119879

119901A120576119901+ 120576119879

119901B120576119887+ 120576119879

119887B120576119901

+120576119879

119887D120576119887+ 120576119879

119904E120576119904minus 120576119879

119887Nth

minus 120576119879

119887Mth

119889Ω

(9)



119880 (120575) =

1

2

120575119879K120575 (10)


119879 (120575) =

1

2

int

Ω

119901 (2

119900+ V2119900+ 2

119900) + 119868 (

1205792

119909+

1205792

119910) 119889Ω (11)

where = int

ℎ2

minusℎ2

120588(119911)119889119911 119868 = int

ℎ2

minusℎ2

1199112




119881 (120575) = int

Ω

1

2

[119873th1199091199091199082

119909+ 119873

th1199101199101199082

119910+ 2119873

th119909119910119908119909119908119910]

+

ℎ2

24

[119873th119909119909

(1205792

119909119909+ 1205792

119910119909) + 119873

2

119910119910(1205792

119909119910+ 1205792

119910119910)

+ 2119873th119909119910

(120579119909119909

120579119909119910

+ 120579119910119909

120579119910119910

)] 119889Ω

(12)


Static bending

K120575 = F (13)

Free vibration

M 120575 + (K + K

119866) 120575 = 0 (14)

Buckling analysis


(K + 120582119872K119866) 120575 = 0 (15)

Thermal buckling

(K + 120582119879K119866) 120575 = 0 (16)

where 120582119872






119894(119894 = 119888119898)



119881119888(119911) = (

2119911 + ℎ

2ℎ

)

119899

(17)




119870eff minus 119870119898

119870119888minus 119870119898

=

119881119888

1 + 119881119898(3 (119870119888minus 119870119898) (3119870

119898+ 4119866119898))

119866eff minus 119866119898

119866119888minus 119866119898

=

119881119888

1 + 119881119898((119866119888minus 119866119898) (119866119898+ 1198911))

(18)

where

1198911=

119866119898(9119870119898+ 8119866119898)

6 (119870119898+ 2119866119898)

(19)


119864eff =

9119870eff119866eff3119870eff + 119866eff

]eff =


(20)




120581eff minus 120581119898

120581119888minus 120581119898

=

119881119888

1 + 119881119898((120581119888minus 120581119898) 3120581119898)

120572eff minus 120572119898

120572119888minus 120572119898

=

(1119870eff minus 1119870119898)

(1119870119888minus 1119870

119898)

(21)


119875 = 119875119900(119875minus1119879minus1

+ 1 + 1198751119879 + 11987521198792

+ 11987531198793

) (22)

where 119875119900 119875minus1 1198751 1198752 and 119875




minus

119889

119889119911

[120581 (119911)

119889119879

119889119911

] = 0 119879 =

119879119888

at 119911 =

ℎ

2

119879119898

at 119911 = minus

ℎ

2

(23)


119879 (119911) = 119879119898+ (119879119888minus 119879119898) 120578 (119911 ℎ) (24)


1

2

3

O

Δ1

Δ2

Δ3




where

120578 (119911 ℎ) =

1

119862

[(

2119911 + ℎ

2ℎ

) minus

120581119888119898

(119899 + 1) 120581119898

(

2119911 + ℎ

2ℎ

)

119899+1

+

1205812

119888119898

(2119899 + 1) 1205812

119898

(

2119911 + ℎ

2ℎ

)

2119899+1

minus

1205813

119888119898

(3119899 + 1) 1205813

119898

(

2119911 + ℎ

2ℎ

)

3119899+1

+

1205814

119888119898

(4119899 + 1) 1205814

119898

(

2119911 + ℎ

2ℎ

)

4119899+1

minus

1205815

119888119898

(5119899 + 1) 1205815

119898

(

2119911 + ℎ

2ℎ

)

5119899+1

]

(25)

119862 = 1 minus

120581119888119898

(119899 + 1) 120581119898

+

1205812

119888119898

(2119899 + 1) 1205812

119898

minus

1205813

119888119898

(3119899 + 1) 1205813

119898

+

1205814

119888119898

(4119899 + 1) 1205814

119898

minus

1205815

119888119898

(5119899 + 1) 1205815

119898

(26)

where 120581119888119898

= 120581119888minus 120581119898



119909 120579119910 is employed


uℎ = sum

119868

119873119868120575119868 (27)




119873 = [1 minus 120585 minus 120578 120578 120585] (28)


119890as


3such that Ω

119890= ⋃3



(119909119900 119910119900) =

1

3

(119909119868 119910119868) (29)


120575119890119874

=

1

3

120575119890119868 (30)


1are given by

120576119901= [p11

p12

p13

]

120575119890119874

1205751198901

1205751198902

120576119887= [b11

b12

b13

]

120575119890119874

1205751198901

1205751198902

120576119904= [s11

s12

s13

]

120575119890119874

1205751198901

1205751198902

(31)


in (31) we obtain

1205761

119901= [

1

3

p11

+ p12

1

3

p11

+ p13

1

3

p11

]

times

1205751198901

1205751198902

1205751198903

= B1119901120575119890

1205761

119887= [

1

3

b11

+ b12

1

3

b11

+ b13

1

3

b11

]

times

1205751198901

1205751198902

1205751198903

= B1119887120575119890


1205761

119904= [

1

3

s11

+ s12

1

3

s11

+ s13

1

3

s11

]

times

1205751198901

1205751198902

1205751198903

= B1119904120575119890

(32)

where p119894 (119894 = 1 2 3) b

119894 (119894 = 1 2 3) and s

119894 (119894 = 1 2 3) are

given by

B119901=

1

2119860119890

[

[

[

[

119887 minus 119888 0 0 0 0

0 119889 minus 119886 0 0 0

119889 minus 119886 119887 minus 119888 0 0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

p1

119888 0 0 0 0

0 minus119889 0 0 0

minus119889 119888 0 0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

p2

minus119887 0 0 0 0

119886 0 0 0 0

119886 minus119887 0 0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

p3

]

]

]

]

B119887=

1

2119860119890

[

[

[

[

0 0 0 119887 minus 119888 0

0 0 0 0 119889 minus 119886

0 0 0 119889 minus 119886 119887 minus 119888⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

b1

0 0 0 119888 0

0 0 0 0 minus119889

0 0 0 minus119889 119888⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

b2

0 0 0 minus119887 0

0 0 0 0 119886

0 0 0 119886 minus119887⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

b3

]

]

]

]

B119904=

1

2119860119890

[

[

0 0 119887 minus 119888 119860119890

0

0 0 119889 minus 119886 0 119860119890⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

s1

0 0 119888 1198861198882 1198871198882


s2


0 0 119886 1198861198892 1198861198882⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

s3

]

]

(33)

where 119886 = 1199092minus 1199091 119887 = 119910

2minus 1199101 119888 = 119910

3minus 1199101 119889 = 119909

3minus 1199091





119887


119890as

follows

120576119901= int

Ω119890

120576119887Φ119890(x) 119889Ω =

3

sum

119894=1

120576119894

119901int

119894

Φ119890(x) 119889Ω

120576119887= int

Ω119890

120576119887Φ119890(x) 119889Ω =

3

sum

119894=1

120576119894

119887int

119894

Φ119890(x) 119889Ω

120576119904= int

Ω119890

120576119904Φ119890(x) 119889Ω =

3

sum

119894=1

120576119894

119904int

119894

Φ119890(x) 119889Ω

(34)


Φ (x) =

1

119860119888

x isin Ω119888

0 x notin Ω119888

(35)



120576119901 120576119887 120576119904 =

sum3

119894=1119860119894

120576119894

119901 120576119894

119887 120576119894

119904

119860119890

(36)


K = int

Ω119890



(37)




ShearFac =

ℎ3

ℎ2+ 120572ℎ2

119890

(38)






b

b

c

d

120578

1

120585

2

3



119906119900= 119908119900= 120579119910= 0 on 119909 = 0 119886

V119900= 119908119900= 120579119909= 0 on 119910 = 0 119887

(39)


119906119900= 119908119900= 120579119910= V119900= 120579119909= 0 on 119909 = 0 119886 119910 = 0 119887 (40)


119900 V1015840119900 1199081015840

119900)


1199101015840

1199111015840


120575 = L1198921205751015840

(41)


L119892=

[

[

[

[

[

[

cos120595 sin120595 0 0 0

minus sin120595 cos120595 0 0 0

0 0 1 0 0

0 0 0 cos120595 sin120595

0 0 0 minus sin120595 cos120595

]

]

]

]

]

]

(42)





119888(119864119888ℎ3

12(1 minus





is 119864119888





119888= 100119908

119888(119864119898ℎ3



120596 = 1205961198862

radic

120588119888ℎ

119863119888

(43)


12(1 minus




3N4SUS304 are


119888= 2370 kgm3and 120581

119888= 919WmK




119875minus1

1198751

1198752

1198753

Si3N4119864 (Pa) 34843119890

9 00 minus3070119890minus4

2160119890minus7


120572 (1K) 58723119890minus6 00 9095119890

minus4 00 00

SUS304 119864 (Pa) 201041198909 00 3079119890

minus4


120572 (1K) 12330119890minus6 00 8086119890

minus4 00 00

002

004

006

008

01

012

014

016

018

Nor

mal

ized

cent

ral d

eflec

tion


ah

100 101 102 103 104


for Si3N4and 120588

119898= 8166 kgm3and 120581

119898= 1204WmK for


= 119879119898


119898=


a

b

r





2) The


2is 119864119888

= 151 GPa and for Alis 119864119898




119879119888 119879119898


300 300 Present 183570 110690 90260 85880[48] 183731 110288 90128 85870

400 300 Present 179778 107979 88626 83182[48] 179620 107860 87530 83090

600 300 Present 171205 101679 81253 76516[48] 171050 101550 81150 76420


[1205962

120588119888ℎ1198864

119863119888(1 minus ]2)]

14



0 005 01 015 02 025 03 035 045

10

15

20

25

30

35

40

Non

dim

ensio

naliz

ed m

ode 1

freq

uenc

y

ra

n = 0n = 5


119888= 400K 119879

119898=



120582cru =

1198730

119909119909cr1198872

1205872119863119888

120582crb =

1198730

119910119910cr1198872

1205872119863119888

(44)

05 1 15 2 251

2

3

4

5

6

7

ab

n = 0

n = 5

Criti

cal b

uckl

ing

load

120582cr

u



119888= 119864119888ℎ3




119879119888


300 [49] 176855 106681 96040 87113 82850Present 177122 106845 96188 87246 82976

400 [37] 174690 105174 94618 85738 81484Present 175488 105775 95197 86309 82059


Skew angle 120582cr



0∘ 120582cru 40010 40034 17956 18052 12624 10846120582crb 20002 20017 08980 09028 06312 05423

15∘ 120582cru 43946 44007 19716 19799 13859 11915120582crb 21154 21187 09517 09561 06683 05741

30∘ 120582cru 58966 59317 26496 26496 18586 16020120582crb 25365 25491 11519 11520 08047 06909


119873119900

119909119909cr 1198872

1205872

119863119898





0 005 01 015 02 025 03 035 040

2

4

6

8

10

12

14

Criti

cal b

uckl

ing

load

120582un

i

ra




119873119900

119909119909cr1198872

1205872




119888=

380GPa 120581119888=104WmK and 120572

119888= 74 times 10


119898= 70GPa 120581

119898= 204WmK and 120572

119898= 23 times








05 1 15 2 25500

1000

1500

2000

2500

3000

3500

LinearNonlinear

ab

Criti

cal b

uckl

ing

tem

pera

ture

ΔT

cr



0 5 10 15 20 25 301000

1500

2000

2500

3000

3500

4000

4500

Skew angle 120595


Criti

cal b

uckl

ing

tem

pera

ture

ΔT

cr




5 Conclusion



0 005 01 015 02 025 03 035 04500

1000

1500

2000

2500

3000

3500

Criti

cal b

uckl

ing

tem

pera

ture

ra

n = 0






Acknowledgment


References





























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










x

yz

a

b

h

(a)

a

b

y

120595

y998400

x x998400

(b)





N =

119873119909119909

119873119910119910

119873119909119910

= A120576119901+ B120576119887minus Nth

M =

119872119909119909

119872119910119910

119872119909119910

= B120576119901+D119887120576119887minusMth

(4)


119894119895 and D

119887= 119863119894119895 (119894 119895 =


119860119894119895 119861119894119895 119863119894119895 = int

ℎ2

minusℎ2

1198761198941198951 119911 119911

2

119889119911 (5)



equation

119876119894119895= 119864119894119895120576119904 (6)

whereE = 119864119894119895= int

ℎ2

minusℎ2





as

11987611=11987622

=

119864 (119911)

1 minus ]2 119876

12=

]119864 (119911)

1 minus ]2 119876

16= 11987626

= 0

11987644

= 11987655

= 11987666

=

119864 (119911)

2 (1 + ])

(7)


Nth=

119873th119909119909

119873th119910119910

119873th119909119910

= int

ℎ2

minusℎ2

119876119894119895120572 (119911 119879)

1

1

0

119879 (119911) 119889119911

Mth=

119872th119909119909

119872th119910119910

119872th119909119910

= int

ℎ2

minusℎ2

119876119894119895120572 (119911 119879)

1

1

0

119879 (119911) 119911 119889119911

(8)





119880 (120575) =

1

2

int

Ω

120576119879

119901A120576119901+ 120576119879

119901B120576119887+ 120576119879

119887B120576119901

+120576119879

119887D120576119887+ 120576119879

119904E120576119904minus 120576119879

119887Nth

minus 120576119879

119887Mth

119889Ω

(9)



119880 (120575) =

1

2

120575119879K120575 (10)


119879 (120575) =

1

2

int

Ω

119901 (2

119900+ V2119900+ 2

119900) + 119868 (

1205792

119909+

1205792

119910) 119889Ω (11)

where = int

ℎ2

minusℎ2

120588(119911)119889119911 119868 = int

ℎ2

minusℎ2

1199112




119881 (120575) = int

Ω

1

2

[119873th1199091199091199082

119909+ 119873

th1199101199101199082

119910+ 2119873

th119909119910119908119909119908119910]

+

ℎ2

24

[119873th119909119909

(1205792

119909119909+ 1205792

119910119909) + 119873

2

119910119910(1205792

119909119910+ 1205792

119910119910)

+ 2119873th119909119910

(120579119909119909

120579119909119910

+ 120579119910119909

120579119910119910

)] 119889Ω

(12)


Static bending

K120575 = F (13)

Free vibration

M 120575 + (K + K

119866) 120575 = 0 (14)

Buckling analysis


(K + 120582119872K119866) 120575 = 0 (15)

Thermal buckling

(K + 120582119879K119866) 120575 = 0 (16)

where 120582119872






119894(119894 = 119888119898)



119881119888(119911) = (

2119911 + ℎ

2ℎ

)

119899

(17)




119870eff minus 119870119898

119870119888minus 119870119898

=

119881119888

1 + 119881119898(3 (119870119888minus 119870119898) (3119870

119898+ 4119866119898))

119866eff minus 119866119898

119866119888minus 119866119898

=

119881119888

1 + 119881119898((119866119888minus 119866119898) (119866119898+ 1198911))

(18)

where

1198911=

119866119898(9119870119898+ 8119866119898)

6 (119870119898+ 2119866119898)

(19)


119864eff =

9119870eff119866eff3119870eff + 119866eff

]eff =


(20)




120581eff minus 120581119898

120581119888minus 120581119898

=

119881119888

1 + 119881119898((120581119888minus 120581119898) 3120581119898)

120572eff minus 120572119898

120572119888minus 120572119898

=

(1119870eff minus 1119870119898)

(1119870119888minus 1119870

119898)

(21)


119875 = 119875119900(119875minus1119879minus1

+ 1 + 1198751119879 + 11987521198792

+ 11987531198793

) (22)

where 119875119900 119875minus1 1198751 1198752 and 119875




minus

119889

119889119911

[120581 (119911)

119889119879

119889119911

] = 0 119879 =

119879119888

at 119911 =

ℎ

2

119879119898

at 119911 = minus

ℎ

2

(23)


119879 (119911) = 119879119898+ (119879119888minus 119879119898) 120578 (119911 ℎ) (24)


1

2

3

O

Δ1

Δ2

Δ3




where

120578 (119911 ℎ) =

1

119862

[(

2119911 + ℎ

2ℎ

) minus

120581119888119898

(119899 + 1) 120581119898

(

2119911 + ℎ

2ℎ

)

119899+1

+

1205812

119888119898

(2119899 + 1) 1205812

119898

(

2119911 + ℎ

2ℎ

)

2119899+1

minus

1205813

119888119898

(3119899 + 1) 1205813

119898

(

2119911 + ℎ

2ℎ

)

3119899+1

+

1205814

119888119898

(4119899 + 1) 1205814

119898

(

2119911 + ℎ

2ℎ

)

4119899+1

minus

1205815

119888119898

(5119899 + 1) 1205815

119898

(

2119911 + ℎ

2ℎ

)

5119899+1

]

(25)

119862 = 1 minus

120581119888119898

(119899 + 1) 120581119898

+

1205812

119888119898

(2119899 + 1) 1205812

119898

minus

1205813

119888119898

(3119899 + 1) 1205813

119898

+

1205814

119888119898

(4119899 + 1) 1205814

119898

minus

1205815

119888119898

(5119899 + 1) 1205815

119898

(26)

where 120581119888119898

= 120581119888minus 120581119898



119909 120579119910 is employed


uℎ = sum

119868

119873119868120575119868 (27)




119873 = [1 minus 120585 minus 120578 120578 120585] (28)


119890as


3such that Ω

119890= ⋃3



(119909119900 119910119900) =

1

3

(119909119868 119910119868) (29)


120575119890119874

=

1

3

120575119890119868 (30)


1are given by

120576119901= [p11

p12

p13

]

120575119890119874

1205751198901

1205751198902

120576119887= [b11

b12

b13

]

120575119890119874

1205751198901

1205751198902

120576119904= [s11

s12

s13

]

120575119890119874

1205751198901

1205751198902

(31)


in (31) we obtain

1205761

119901= [

1

3

p11

+ p12

1

3

p11

+ p13

1

3

p11

]

times

1205751198901

1205751198902

1205751198903

= B1119901120575119890

1205761

119887= [

1

3

b11

+ b12

1

3

b11

+ b13

1

3

b11

]

times

1205751198901

1205751198902

1205751198903

= B1119887120575119890


1205761

119904= [

1

3

s11

+ s12

1

3

s11

+ s13

1

3

s11

]

times

1205751198901

1205751198902

1205751198903

= B1119904120575119890

(32)

where p119894 (119894 = 1 2 3) b

119894 (119894 = 1 2 3) and s

119894 (119894 = 1 2 3) are

given by

B119901=

1

2119860119890

[

[

[

[

119887 minus 119888 0 0 0 0

0 119889 minus 119886 0 0 0

119889 minus 119886 119887 minus 119888 0 0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

p1

119888 0 0 0 0

0 minus119889 0 0 0

minus119889 119888 0 0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

p2

minus119887 0 0 0 0

119886 0 0 0 0

119886 minus119887 0 0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

p3

]

]

]

]

B119887=

1

2119860119890

[

[

[

[

0 0 0 119887 minus 119888 0

0 0 0 0 119889 minus 119886

0 0 0 119889 minus 119886 119887 minus 119888⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

b1

0 0 0 119888 0

0 0 0 0 minus119889

0 0 0 minus119889 119888⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

b2

0 0 0 minus119887 0

0 0 0 0 119886

0 0 0 119886 minus119887⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

b3

]

]

]

]

B119904=

1

2119860119890

[

[

0 0 119887 minus 119888 119860119890

0

0 0 119889 minus 119886 0 119860119890⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

s1

0 0 119888 1198861198882 1198871198882


s2


0 0 119886 1198861198892 1198861198882⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

s3

]

]

(33)

where 119886 = 1199092minus 1199091 119887 = 119910

2minus 1199101 119888 = 119910

3minus 1199101 119889 = 119909

3minus 1199091





119887


119890as

follows

120576119901= int

Ω119890

120576119887Φ119890(x) 119889Ω =

3

sum

119894=1

120576119894

119901int

119894

Φ119890(x) 119889Ω

120576119887= int

Ω119890

120576119887Φ119890(x) 119889Ω =

3

sum

119894=1

120576119894

119887int

119894

Φ119890(x) 119889Ω

120576119904= int

Ω119890

120576119904Φ119890(x) 119889Ω =

3

sum

119894=1

120576119894

119904int

119894

Φ119890(x) 119889Ω

(34)


Φ (x) =

1

119860119888

x isin Ω119888

0 x notin Ω119888

(35)



120576119901 120576119887 120576119904 =

sum3

119894=1119860119894

120576119894

119901 120576119894

119887 120576119894

119904

119860119890

(36)


K = int

Ω119890



(37)




ShearFac =

ℎ3

ℎ2+ 120572ℎ2

119890

(38)






b

b

c

d

120578

1

120585

2

3



119906119900= 119908119900= 120579119910= 0 on 119909 = 0 119886

V119900= 119908119900= 120579119909= 0 on 119910 = 0 119887

(39)


119906119900= 119908119900= 120579119910= V119900= 120579119909= 0 on 119909 = 0 119886 119910 = 0 119887 (40)


119900 V1015840119900 1199081015840

119900)


1199101015840

1199111015840


120575 = L1198921205751015840

(41)


L119892=

[

[

[

[

[

[

cos120595 sin120595 0 0 0

minus sin120595 cos120595 0 0 0

0 0 1 0 0

0 0 0 cos120595 sin120595

0 0 0 minus sin120595 cos120595

]

]

]

]

]

]

(42)





119888(119864119888ℎ3

12(1 minus





is 119864119888





119888= 100119908

119888(119864119898ℎ3



120596 = 1205961198862

radic

120588119888ℎ

119863119888

(43)


12(1 minus




3N4SUS304 are


119888= 2370 kgm3and 120581

119888= 919WmK




119875minus1

1198751

1198752

1198753

Si3N4119864 (Pa) 34843119890

9 00 minus3070119890minus4

2160119890minus7


120572 (1K) 58723119890minus6 00 9095119890

minus4 00 00

SUS304 119864 (Pa) 201041198909 00 3079119890

minus4


120572 (1K) 12330119890minus6 00 8086119890

minus4 00 00

002

004

006

008

01

012

014

016

018

Nor

mal

ized

cent

ral d

eflec

tion


ah

100 101 102 103 104


for Si3N4and 120588

119898= 8166 kgm3and 120581

119898= 1204WmK for


= 119879119898


119898=


a

b

r





2) The


2is 119864119888

= 151 GPa and for Alis 119864119898




119879119888 119879119898


300 300 Present 183570 110690 90260 85880[48] 183731 110288 90128 85870

400 300 Present 179778 107979 88626 83182[48] 179620 107860 87530 83090

600 300 Present 171205 101679 81253 76516[48] 171050 101550 81150 76420


[1205962

120588119888ℎ1198864

119863119888(1 minus ]2)]

14



0 005 01 015 02 025 03 035 045

10

15

20

25

30

35

40

Non

dim

ensio

naliz

ed m

ode 1

freq

uenc

y

ra

n = 0n = 5


119888= 400K 119879

119898=



120582cru =

1198730

119909119909cr1198872

1205872119863119888

120582crb =

1198730

119910119910cr1198872

1205872119863119888

(44)

05 1 15 2 251

2

3

4

5

6

7

ab

n = 0

n = 5

Criti

cal b

uckl

ing

load

120582cr

u



119888= 119864119888ℎ3




119879119888


300 [49] 176855 106681 96040 87113 82850Present 177122 106845 96188 87246 82976

400 [37] 174690 105174 94618 85738 81484Present 175488 105775 95197 86309 82059


Skew angle 120582cr



0∘ 120582cru 40010 40034 17956 18052 12624 10846120582crb 20002 20017 08980 09028 06312 05423

15∘ 120582cru 43946 44007 19716 19799 13859 11915120582crb 21154 21187 09517 09561 06683 05741

30∘ 120582cru 58966 59317 26496 26496 18586 16020120582crb 25365 25491 11519 11520 08047 06909


119873119900

119909119909cr 1198872

1205872

119863119898





0 005 01 015 02 025 03 035 040

2

4

6

8

10

12

14

Criti

cal b

uckl

ing

load

120582un

i

ra




119873119900

119909119909cr1198872

1205872




119888=

380GPa 120581119888=104WmK and 120572

119888= 74 times 10


119898= 70GPa 120581

119898= 204WmK and 120572

119898= 23 times








05 1 15 2 25500

1000

1500

2000

2500

3000

3500

LinearNonlinear

ab

Criti

cal b

uckl

ing

tem

pera

ture

ΔT

cr



0 5 10 15 20 25 301000

1500

2000

2500

3000

3500

4000

4500

Skew angle 120595


Criti

cal b

uckl

ing

tem

pera

ture

ΔT

cr




5 Conclusion



0 005 01 015 02 025 03 035 04500

1000

1500

2000

2500

3000

3500

Criti

cal b

uckl

ing

tem

pera

ture

ra

n = 0






Acknowledgment


References





























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119881 (120575) = int

Ω

1

2

[119873th1199091199091199082

119909+ 119873

th1199101199101199082

119910+ 2119873

th119909119910119908119909119908119910]

+

ℎ2

24

[119873th119909119909

(1205792

119909119909+ 1205792

119910119909) + 119873

2

119910119910(1205792

119909119910+ 1205792

119910119910)

+ 2119873th119909119910

(120579119909119909

120579119909119910

+ 120579119910119909

120579119910119910

)] 119889Ω

(12)


Static bending

K120575 = F (13)

Free vibration

M 120575 + (K + K

119866) 120575 = 0 (14)

Buckling analysis


(K + 120582119872K119866) 120575 = 0 (15)

Thermal buckling

(K + 120582119879K119866) 120575 = 0 (16)

where 120582119872






119894(119894 = 119888119898)



119881119888(119911) = (

2119911 + ℎ

2ℎ

)

119899

(17)




119870eff minus 119870119898

119870119888minus 119870119898

=

119881119888

1 + 119881119898(3 (119870119888minus 119870119898) (3119870

119898+ 4119866119898))

119866eff minus 119866119898

119866119888minus 119866119898

=

119881119888

1 + 119881119898((119866119888minus 119866119898) (119866119898+ 1198911))

(18)

where

1198911=

119866119898(9119870119898+ 8119866119898)

6 (119870119898+ 2119866119898)

(19)


119864eff =

9119870eff119866eff3119870eff + 119866eff

]eff =


(20)




120581eff minus 120581119898

120581119888minus 120581119898

=

119881119888

1 + 119881119898((120581119888minus 120581119898) 3120581119898)

120572eff minus 120572119898

120572119888minus 120572119898

=

(1119870eff minus 1119870119898)

(1119870119888minus 1119870

119898)

(21)


119875 = 119875119900(119875minus1119879minus1

+ 1 + 1198751119879 + 11987521198792

+ 11987531198793

) (22)

where 119875119900 119875minus1 1198751 1198752 and 119875




minus

119889

119889119911

[120581 (119911)

119889119879

119889119911

] = 0 119879 =

119879119888

at 119911 =

ℎ

2

119879119898

at 119911 = minus

ℎ

2

(23)


119879 (119911) = 119879119898+ (119879119888minus 119879119898) 120578 (119911 ℎ) (24)


1

2

3

O

Δ1

Δ2

Δ3




where

120578 (119911 ℎ) =

1

119862

[(

2119911 + ℎ

2ℎ

) minus

120581119888119898

(119899 + 1) 120581119898

(

2119911 + ℎ

2ℎ

)

119899+1

+

1205812

119888119898

(2119899 + 1) 1205812

119898

(

2119911 + ℎ

2ℎ

)

2119899+1

minus

1205813

119888119898

(3119899 + 1) 1205813

119898

(

2119911 + ℎ

2ℎ

)

3119899+1

+

1205814

119888119898

(4119899 + 1) 1205814

119898

(

2119911 + ℎ

2ℎ

)

4119899+1

minus

1205815

119888119898

(5119899 + 1) 1205815

119898

(

2119911 + ℎ

2ℎ

)

5119899+1

]

(25)

119862 = 1 minus

120581119888119898

(119899 + 1) 120581119898

+

1205812

119888119898

(2119899 + 1) 1205812

119898

minus

1205813

119888119898

(3119899 + 1) 1205813

119898

+

1205814

119888119898

(4119899 + 1) 1205814

119898

minus

1205815

119888119898

(5119899 + 1) 1205815

119898

(26)

where 120581119888119898

= 120581119888minus 120581119898



119909 120579119910 is employed


uℎ = sum

119868

119873119868120575119868 (27)




119873 = [1 minus 120585 minus 120578 120578 120585] (28)


119890as


3such that Ω

119890= ⋃3



(119909119900 119910119900) =

1

3

(119909119868 119910119868) (29)


120575119890119874

=

1

3

120575119890119868 (30)


1are given by

120576119901= [p11

p12

p13

]

120575119890119874

1205751198901

1205751198902

120576119887= [b11

b12

b13

]

120575119890119874

1205751198901

1205751198902

120576119904= [s11

s12

s13

]

120575119890119874

1205751198901

1205751198902

(31)


in (31) we obtain

1205761

119901= [

1

3

p11

+ p12

1

3

p11

+ p13

1

3

p11

]

times

1205751198901

1205751198902

1205751198903

= B1119901120575119890

1205761

119887= [

1

3

b11

+ b12

1

3

b11

+ b13

1

3

b11

]

times

1205751198901

1205751198902

1205751198903

= B1119887120575119890


1205761

119904= [

1

3

s11

+ s12

1

3

s11

+ s13

1

3

s11

]

times

1205751198901

1205751198902

1205751198903

= B1119904120575119890

(32)

where p119894 (119894 = 1 2 3) b

119894 (119894 = 1 2 3) and s

119894 (119894 = 1 2 3) are

given by

B119901=

1

2119860119890

[

[

[

[

119887 minus 119888 0 0 0 0

0 119889 minus 119886 0 0 0

119889 minus 119886 119887 minus 119888 0 0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

p1

119888 0 0 0 0

0 minus119889 0 0 0

minus119889 119888 0 0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

p2

minus119887 0 0 0 0

119886 0 0 0 0

119886 minus119887 0 0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

p3

]

]

]

]

B119887=

1

2119860119890

[

[

[

[

0 0 0 119887 minus 119888 0

0 0 0 0 119889 minus 119886

0 0 0 119889 minus 119886 119887 minus 119888⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

b1

0 0 0 119888 0

0 0 0 0 minus119889

0 0 0 minus119889 119888⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

b2

0 0 0 minus119887 0

0 0 0 0 119886

0 0 0 119886 minus119887⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

b3

]

]

]

]

B119904=

1

2119860119890

[

[

0 0 119887 minus 119888 119860119890

0

0 0 119889 minus 119886 0 119860119890⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

s1

0 0 119888 1198861198882 1198871198882


s2


0 0 119886 1198861198892 1198861198882⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

s3

]

]

(33)

where 119886 = 1199092minus 1199091 119887 = 119910

2minus 1199101 119888 = 119910

3minus 1199101 119889 = 119909

3minus 1199091





119887


119890as

follows

120576119901= int

Ω119890

120576119887Φ119890(x) 119889Ω =

3

sum

119894=1

120576119894

119901int

119894

Φ119890(x) 119889Ω

120576119887= int

Ω119890

120576119887Φ119890(x) 119889Ω =

3

sum

119894=1

120576119894

119887int

119894

Φ119890(x) 119889Ω

120576119904= int

Ω119890

120576119904Φ119890(x) 119889Ω =

3

sum

119894=1

120576119894

119904int

119894

Φ119890(x) 119889Ω

(34)


Φ (x) =

1

119860119888

x isin Ω119888

0 x notin Ω119888

(35)



120576119901 120576119887 120576119904 =

sum3

119894=1119860119894

120576119894

119901 120576119894

119887 120576119894

119904

119860119890

(36)


K = int

Ω119890



(37)




ShearFac =

ℎ3

ℎ2+ 120572ℎ2

119890

(38)






b

b

c

d

120578

1

120585

2

3



119906119900= 119908119900= 120579119910= 0 on 119909 = 0 119886

V119900= 119908119900= 120579119909= 0 on 119910 = 0 119887

(39)


119906119900= 119908119900= 120579119910= V119900= 120579119909= 0 on 119909 = 0 119886 119910 = 0 119887 (40)


119900 V1015840119900 1199081015840

119900)


1199101015840

1199111015840


120575 = L1198921205751015840

(41)


L119892=

[

[

[

[

[

[

cos120595 sin120595 0 0 0

minus sin120595 cos120595 0 0 0

0 0 1 0 0

0 0 0 cos120595 sin120595

0 0 0 minus sin120595 cos120595

]

]

]

]

]

]

(42)





119888(119864119888ℎ3

12(1 minus





is 119864119888





119888= 100119908

119888(119864119898ℎ3



120596 = 1205961198862

radic

120588119888ℎ

119863119888

(43)


12(1 minus




3N4SUS304 are


119888= 2370 kgm3and 120581

119888= 919WmK




119875minus1

1198751

1198752

1198753

Si3N4119864 (Pa) 34843119890

9 00 minus3070119890minus4

2160119890minus7


120572 (1K) 58723119890minus6 00 9095119890

minus4 00 00

SUS304 119864 (Pa) 201041198909 00 3079119890

minus4


120572 (1K) 12330119890minus6 00 8086119890

minus4 00 00

002

004

006

008

01

012

014

016

018

Nor

mal

ized

cent

ral d

eflec

tion


ah

100 101 102 103 104


for Si3N4and 120588

119898= 8166 kgm3and 120581

119898= 1204WmK for


= 119879119898


119898=


a

b

r





2) The


2is 119864119888

= 151 GPa and for Alis 119864119898




119879119888 119879119898


300 300 Present 183570 110690 90260 85880[48] 183731 110288 90128 85870

400 300 Present 179778 107979 88626 83182[48] 179620 107860 87530 83090

600 300 Present 171205 101679 81253 76516[48] 171050 101550 81150 76420


[1205962

120588119888ℎ1198864

119863119888(1 minus ]2)]

14



0 005 01 015 02 025 03 035 045

10

15

20

25

30

35

40

Non

dim

ensio

naliz

ed m

ode 1

freq

uenc

y

ra

n = 0n = 5


119888= 400K 119879

119898=



120582cru =

1198730

119909119909cr1198872

1205872119863119888

120582crb =

1198730

119910119910cr1198872

1205872119863119888

(44)

05 1 15 2 251

2

3

4

5

6

7

ab

n = 0

n = 5

Criti

cal b

uckl

ing

load

120582cr

u



119888= 119864119888ℎ3




119879119888


300 [49] 176855 106681 96040 87113 82850Present 177122 106845 96188 87246 82976

400 [37] 174690 105174 94618 85738 81484Present 175488 105775 95197 86309 82059


Skew angle 120582cr



0∘ 120582cru 40010 40034 17956 18052 12624 10846120582crb 20002 20017 08980 09028 06312 05423

15∘ 120582cru 43946 44007 19716 19799 13859 11915120582crb 21154 21187 09517 09561 06683 05741

30∘ 120582cru 58966 59317 26496 26496 18586 16020120582crb 25365 25491 11519 11520 08047 06909


119873119900

119909119909cr 1198872

1205872

119863119898





0 005 01 015 02 025 03 035 040

2

4

6

8

10

12

14

Criti

cal b

uckl

ing

load

120582un

i

ra




119873119900

119909119909cr1198872

1205872




119888=

380GPa 120581119888=104WmK and 120572

119888= 74 times 10


119898= 70GPa 120581

119898= 204WmK and 120572

119898= 23 times








05 1 15 2 25500

1000

1500

2000

2500

3000

3500

LinearNonlinear

ab

Criti

cal b

uckl

ing

tem

pera

ture

ΔT

cr



0 5 10 15 20 25 301000

1500

2000

2500

3000

3500

4000

4500

Skew angle 120595


Criti

cal b

uckl

ing

tem

pera

ture

ΔT

cr




5 Conclusion



0 005 01 015 02 025 03 035 04500

1000

1500

2000

2500

3000

3500

Criti

cal b

uckl

ing

tem

pera

ture

ra

n = 0






Acknowledgment


References





























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










1

2

3

O

Δ1

Δ2

Δ3




where

120578 (119911 ℎ) =

1

119862

[(

2119911 + ℎ

2ℎ

) minus

120581119888119898

(119899 + 1) 120581119898

(

2119911 + ℎ

2ℎ

)

119899+1

+

1205812

119888119898

(2119899 + 1) 1205812

119898

(

2119911 + ℎ

2ℎ

)

2119899+1

minus

1205813

119888119898

(3119899 + 1) 1205813

119898

(

2119911 + ℎ

2ℎ

)

3119899+1

+

1205814

119888119898

(4119899 + 1) 1205814

119898

(

2119911 + ℎ

2ℎ

)

4119899+1

minus

1205815

119888119898

(5119899 + 1) 1205815

119898

(

2119911 + ℎ

2ℎ

)

5119899+1

]

(25)

119862 = 1 minus

120581119888119898

(119899 + 1) 120581119898

+

1205812

119888119898

(2119899 + 1) 1205812

119898

minus

1205813

119888119898

(3119899 + 1) 1205813

119898

+

1205814

119888119898

(4119899 + 1) 1205814

119898

minus

1205815

119888119898

(5119899 + 1) 1205815

119898

(26)

where 120581119888119898

= 120581119888minus 120581119898



119909 120579119910 is employed


uℎ = sum

119868

119873119868120575119868 (27)




119873 = [1 minus 120585 minus 120578 120578 120585] (28)


119890as


3such that Ω

119890= ⋃3



(119909119900 119910119900) =

1

3

(119909119868 119910119868) (29)


120575119890119874

=

1

3

120575119890119868 (30)


1are given by

120576119901= [p11

p12

p13

]

120575119890119874

1205751198901

1205751198902

120576119887= [b11

b12

b13

]

120575119890119874

1205751198901

1205751198902

120576119904= [s11

s12

s13

]

120575119890119874

1205751198901

1205751198902

(31)


in (31) we obtain

1205761

119901= [

1

3

p11

+ p12

1

3

p11

+ p13

1

3

p11

]

times

1205751198901

1205751198902

1205751198903

= B1119901120575119890

1205761

119887= [

1

3

b11

+ b12

1

3

b11

+ b13

1

3

b11

]

times

1205751198901

1205751198902

1205751198903

= B1119887120575119890


1205761

119904= [

1

3

s11

+ s12

1

3

s11

+ s13

1

3

s11

]

times

1205751198901

1205751198902

1205751198903

= B1119904120575119890

(32)

where p119894 (119894 = 1 2 3) b

119894 (119894 = 1 2 3) and s

119894 (119894 = 1 2 3) are

given by

B119901=

1

2119860119890

[

[

[

[

119887 minus 119888 0 0 0 0

0 119889 minus 119886 0 0 0

119889 minus 119886 119887 minus 119888 0 0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

p1

119888 0 0 0 0

0 minus119889 0 0 0

minus119889 119888 0 0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

p2

minus119887 0 0 0 0

119886 0 0 0 0

119886 minus119887 0 0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

p3

]

]

]

]

B119887=

1

2119860119890

[

[

[

[

0 0 0 119887 minus 119888 0

0 0 0 0 119889 minus 119886

0 0 0 119889 minus 119886 119887 minus 119888⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

b1

0 0 0 119888 0

0 0 0 0 minus119889

0 0 0 minus119889 119888⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

b2

0 0 0 minus119887 0

0 0 0 0 119886

0 0 0 119886 minus119887⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

b3

]

]

]

]

B119904=

1

2119860119890

[

[

0 0 119887 minus 119888 119860119890

0

0 0 119889 minus 119886 0 119860119890⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

s1

0 0 119888 1198861198882 1198871198882


s2


0 0 119886 1198861198892 1198861198882⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

s3

]

]

(33)

where 119886 = 1199092minus 1199091 119887 = 119910

2minus 1199101 119888 = 119910

3minus 1199101 119889 = 119909

3minus 1199091





119887


119890as

follows

120576119901= int

Ω119890

120576119887Φ119890(x) 119889Ω =

3

sum

119894=1

120576119894

119901int

119894

Φ119890(x) 119889Ω

120576119887= int

Ω119890

120576119887Φ119890(x) 119889Ω =

3

sum

119894=1

120576119894

119887int

119894

Φ119890(x) 119889Ω

120576119904= int

Ω119890

120576119904Φ119890(x) 119889Ω =

3

sum

119894=1

120576119894

119904int

119894

Φ119890(x) 119889Ω

(34)


Φ (x) =

1

119860119888

x isin Ω119888

0 x notin Ω119888

(35)



120576119901 120576119887 120576119904 =

sum3

119894=1119860119894

120576119894

119901 120576119894

119887 120576119894

119904

119860119890

(36)


K = int

Ω119890



(37)




ShearFac =

ℎ3

ℎ2+ 120572ℎ2

119890

(38)






b

b

c

d

120578

1

120585

2

3



119906119900= 119908119900= 120579119910= 0 on 119909 = 0 119886

V119900= 119908119900= 120579119909= 0 on 119910 = 0 119887

(39)


119906119900= 119908119900= 120579119910= V119900= 120579119909= 0 on 119909 = 0 119886 119910 = 0 119887 (40)


119900 V1015840119900 1199081015840

119900)


1199101015840

1199111015840


120575 = L1198921205751015840

(41)


L119892=

[

[

[

[

[

[

cos120595 sin120595 0 0 0

minus sin120595 cos120595 0 0 0

0 0 1 0 0

0 0 0 cos120595 sin120595

0 0 0 minus sin120595 cos120595

]

]

]

]

]

]

(42)





119888(119864119888ℎ3

12(1 minus





is 119864119888





119888= 100119908

119888(119864119898ℎ3



120596 = 1205961198862

radic

120588119888ℎ

119863119888

(43)


12(1 minus




3N4SUS304 are


119888= 2370 kgm3and 120581

119888= 919WmK




119875minus1

1198751

1198752

1198753

Si3N4119864 (Pa) 34843119890

9 00 minus3070119890minus4

2160119890minus7


120572 (1K) 58723119890minus6 00 9095119890

minus4 00 00

SUS304 119864 (Pa) 201041198909 00 3079119890

minus4


120572 (1K) 12330119890minus6 00 8086119890

minus4 00 00

002

004

006

008

01

012

014

016

018

Nor

mal

ized

cent

ral d

eflec

tion


ah

100 101 102 103 104


for Si3N4and 120588

119898= 8166 kgm3and 120581

119898= 1204WmK for


= 119879119898


119898=


a

b

r





2) The


2is 119864119888

= 151 GPa and for Alis 119864119898




119879119888 119879119898


300 300 Present 183570 110690 90260 85880[48] 183731 110288 90128 85870

400 300 Present 179778 107979 88626 83182[48] 179620 107860 87530 83090

600 300 Present 171205 101679 81253 76516[48] 171050 101550 81150 76420


[1205962

120588119888ℎ1198864

119863119888(1 minus ]2)]

14



0 005 01 015 02 025 03 035 045

10

15

20

25

30

35

40

Non

dim

ensio

naliz

ed m

ode 1

freq

uenc

y

ra

n = 0n = 5


119888= 400K 119879

119898=



120582cru =

1198730

119909119909cr1198872

1205872119863119888

120582crb =

1198730

119910119910cr1198872

1205872119863119888

(44)

05 1 15 2 251

2

3

4

5

6

7

ab

n = 0

n = 5

Criti

cal b

uckl

ing

load

120582cr

u



119888= 119864119888ℎ3




119879119888


300 [49] 176855 106681 96040 87113 82850Present 177122 106845 96188 87246 82976

400 [37] 174690 105174 94618 85738 81484Present 175488 105775 95197 86309 82059


Skew angle 120582cr



0∘ 120582cru 40010 40034 17956 18052 12624 10846120582crb 20002 20017 08980 09028 06312 05423

15∘ 120582cru 43946 44007 19716 19799 13859 11915120582crb 21154 21187 09517 09561 06683 05741

30∘ 120582cru 58966 59317 26496 26496 18586 16020120582crb 25365 25491 11519 11520 08047 06909


119873119900

119909119909cr 1198872

1205872

119863119898





0 005 01 015 02 025 03 035 040

2

4

6

8

10

12

14

Criti

cal b

uckl

ing

load

120582un

i

ra




119873119900

119909119909cr1198872

1205872




119888=

380GPa 120581119888=104WmK and 120572

119888= 74 times 10


119898= 70GPa 120581

119898= 204WmK and 120572

119898= 23 times








05 1 15 2 25500

1000

1500

2000

2500

3000

3500

LinearNonlinear

ab

Criti

cal b

uckl

ing

tem

pera

ture

ΔT

cr



0 5 10 15 20 25 301000

1500

2000

2500

3000

3500

4000

4500

Skew angle 120595


Criti

cal b

uckl

ing

tem

pera

ture

ΔT

cr




5 Conclusion



0 005 01 015 02 025 03 035 04500

1000

1500

2000

2500

3000

3500

Criti

cal b

uckl

ing

tem

pera

ture

ra

n = 0






Acknowledgment


References





























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










1205761

119904= [

1

3

s11

+ s12

1

3

s11

+ s13

1

3

s11

]

times

1205751198901

1205751198902

1205751198903

= B1119904120575119890

(32)

where p119894 (119894 = 1 2 3) b

119894 (119894 = 1 2 3) and s

119894 (119894 = 1 2 3) are

given by

B119901=

1

2119860119890

[

[

[

[

119887 minus 119888 0 0 0 0

0 119889 minus 119886 0 0 0

119889 minus 119886 119887 minus 119888 0 0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

p1

119888 0 0 0 0

0 minus119889 0 0 0

minus119889 119888 0 0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

p2

minus119887 0 0 0 0

119886 0 0 0 0

119886 minus119887 0 0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

p3

]

]

]

]

B119887=

1

2119860119890

[

[

[

[

0 0 0 119887 minus 119888 0

0 0 0 0 119889 minus 119886

0 0 0 119889 minus 119886 119887 minus 119888⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

b1

0 0 0 119888 0

0 0 0 0 minus119889

0 0 0 minus119889 119888⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

b2

0 0 0 minus119887 0

0 0 0 0 119886

0 0 0 119886 minus119887⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

b3

]

]

]

]

B119904=

1

2119860119890

[

[

0 0 119887 minus 119888 119860119890

0

0 0 119889 minus 119886 0 119860119890⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

s1

0 0 119888 1198861198882 1198871198882


s2


0 0 119886 1198861198892 1198861198882⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

s3

]

]

(33)

where 119886 = 1199092minus 1199091 119887 = 119910

2minus 1199101 119888 = 119910

3minus 1199101 119889 = 119909

3minus 1199091





119887


119890as

follows

120576119901= int

Ω119890

120576119887Φ119890(x) 119889Ω =

3

sum

119894=1

120576119894

119901int

119894

Φ119890(x) 119889Ω

120576119887= int

Ω119890

120576119887Φ119890(x) 119889Ω =

3

sum

119894=1

120576119894

119887int

119894

Φ119890(x) 119889Ω

120576119904= int

Ω119890

120576119904Φ119890(x) 119889Ω =

3

sum

119894=1

120576119894

119904int

119894

Φ119890(x) 119889Ω

(34)


Φ (x) =

1

119860119888

x isin Ω119888

0 x notin Ω119888

(35)



120576119901 120576119887 120576119904 =

sum3

119894=1119860119894

120576119894

119901 120576119894

119887 120576119894

119904

119860119890

(36)


K = int

Ω119890



(37)




ShearFac =

ℎ3

ℎ2+ 120572ℎ2

119890

(38)






b

b

c

d

120578

1

120585

2

3



119906119900= 119908119900= 120579119910= 0 on 119909 = 0 119886

V119900= 119908119900= 120579119909= 0 on 119910 = 0 119887

(39)


119906119900= 119908119900= 120579119910= V119900= 120579119909= 0 on 119909 = 0 119886 119910 = 0 119887 (40)


119900 V1015840119900 1199081015840

119900)


1199101015840

1199111015840


120575 = L1198921205751015840

(41)


L119892=

[

[

[

[

[

[

cos120595 sin120595 0 0 0

minus sin120595 cos120595 0 0 0

0 0 1 0 0

0 0 0 cos120595 sin120595

0 0 0 minus sin120595 cos120595

]

]

]

]

]

]

(42)





119888(119864119888ℎ3

12(1 minus





is 119864119888





119888= 100119908

119888(119864119898ℎ3



120596 = 1205961198862

radic

120588119888ℎ

119863119888

(43)


12(1 minus




3N4SUS304 are


119888= 2370 kgm3and 120581

119888= 919WmK




119875minus1

1198751

1198752

1198753

Si3N4119864 (Pa) 34843119890

9 00 minus3070119890minus4

2160119890minus7


120572 (1K) 58723119890minus6 00 9095119890

minus4 00 00

SUS304 119864 (Pa) 201041198909 00 3079119890

minus4


120572 (1K) 12330119890minus6 00 8086119890

minus4 00 00

002

004

006

008

01

012

014

016

018

Nor

mal

ized

cent

ral d

eflec

tion


ah

100 101 102 103 104


for Si3N4and 120588

119898= 8166 kgm3and 120581

119898= 1204WmK for


= 119879119898


119898=


a

b

r





2) The


2is 119864119888

= 151 GPa and for Alis 119864119898




119879119888 119879119898


300 300 Present 183570 110690 90260 85880[48] 183731 110288 90128 85870

400 300 Present 179778 107979 88626 83182[48] 179620 107860 87530 83090

600 300 Present 171205 101679 81253 76516[48] 171050 101550 81150 76420


[1205962

120588119888ℎ1198864

119863119888(1 minus ]2)]

14



0 005 01 015 02 025 03 035 045

10

15

20

25

30

35

40

Non

dim

ensio

naliz

ed m

ode 1

freq

uenc

y

ra

n = 0n = 5


119888= 400K 119879

119898=



120582cru =

1198730

119909119909cr1198872

1205872119863119888

120582crb =

1198730

119910119910cr1198872

1205872119863119888

(44)

05 1 15 2 251

2

3

4

5

6

7

ab

n = 0

n = 5

Criti

cal b

uckl

ing

load

120582cr

u



119888= 119864119888ℎ3




119879119888


300 [49] 176855 106681 96040 87113 82850Present 177122 106845 96188 87246 82976

400 [37] 174690 105174 94618 85738 81484Present 175488 105775 95197 86309 82059


Skew angle 120582cr



0∘ 120582cru 40010 40034 17956 18052 12624 10846120582crb 20002 20017 08980 09028 06312 05423

15∘ 120582cru 43946 44007 19716 19799 13859 11915120582crb 21154 21187 09517 09561 06683 05741

30∘ 120582cru 58966 59317 26496 26496 18586 16020120582crb 25365 25491 11519 11520 08047 06909


119873119900

119909119909cr 1198872

1205872

119863119898





0 005 01 015 02 025 03 035 040

2

4

6

8

10

12

14

Criti

cal b

uckl

ing

load

120582un

i

ra




119873119900

119909119909cr1198872

1205872




119888=

380GPa 120581119888=104WmK and 120572

119888= 74 times 10


119898= 70GPa 120581

119898= 204WmK and 120572

119898= 23 times








05 1 15 2 25500

1000

1500

2000

2500

3000

3500

LinearNonlinear

ab

Criti

cal b

uckl

ing

tem

pera

ture

ΔT

cr



0 5 10 15 20 25 301000

1500

2000

2500

3000

3500

4000

4500

Skew angle 120595


Criti

cal b

uckl

ing

tem

pera

ture

ΔT

cr




5 Conclusion



0 005 01 015 02 025 03 035 04500

1000

1500

2000

2500

3000

3500

Criti

cal b

uckl

ing

tem

pera

ture

ra

n = 0






Acknowledgment


References





























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










b

b

c

d

120578

1

120585

2

3



119906119900= 119908119900= 120579119910= 0 on 119909 = 0 119886

V119900= 119908119900= 120579119909= 0 on 119910 = 0 119887

(39)


119906119900= 119908119900= 120579119910= V119900= 120579119909= 0 on 119909 = 0 119886 119910 = 0 119887 (40)


119900 V1015840119900 1199081015840

119900)


1199101015840

1199111015840


120575 = L1198921205751015840

(41)


L119892=

[

[

[

[

[

[

cos120595 sin120595 0 0 0

minus sin120595 cos120595 0 0 0

0 0 1 0 0

0 0 0 cos120595 sin120595

0 0 0 minus sin120595 cos120595

]

]

]

]

]

]

(42)





119888(119864119888ℎ3

12(1 minus





is 119864119888





119888= 100119908

119888(119864119898ℎ3



120596 = 1205961198862

radic

120588119888ℎ

119863119888

(43)


12(1 minus




3N4SUS304 are


119888= 2370 kgm3and 120581

119888= 919WmK




119875minus1

1198751

1198752

1198753

Si3N4119864 (Pa) 34843119890

9 00 minus3070119890minus4

2160119890minus7


120572 (1K) 58723119890minus6 00 9095119890

minus4 00 00

SUS304 119864 (Pa) 201041198909 00 3079119890

minus4


120572 (1K) 12330119890minus6 00 8086119890

minus4 00 00

002

004

006

008

01

012

014

016

018

Nor

mal

ized

cent

ral d

eflec

tion


ah

100 101 102 103 104


for Si3N4and 120588

119898= 8166 kgm3and 120581

119898= 1204WmK for


= 119879119898


119898=


a

b

r





2) The


2is 119864119888

= 151 GPa and for Alis 119864119898




119879119888 119879119898


300 300 Present 183570 110690 90260 85880[48] 183731 110288 90128 85870

400 300 Present 179778 107979 88626 83182[48] 179620 107860 87530 83090

600 300 Present 171205 101679 81253 76516[48] 171050 101550 81150 76420


[1205962

120588119888ℎ1198864

119863119888(1 minus ]2)]

14



0 005 01 015 02 025 03 035 045

10

15

20

25

30

35

40

Non

dim

ensio

naliz

ed m

ode 1

freq

uenc

y

ra

n = 0n = 5


119888= 400K 119879

119898=



120582cru =

1198730

119909119909cr1198872

1205872119863119888

120582crb =

1198730

119910119910cr1198872

1205872119863119888

(44)

05 1 15 2 251

2

3

4

5

6

7

ab

n = 0

n = 5

Criti

cal b

uckl

ing

load

120582cr

u



119888= 119864119888ℎ3




119879119888


300 [49] 176855 106681 96040 87113 82850Present 177122 106845 96188 87246 82976

400 [37] 174690 105174 94618 85738 81484Present 175488 105775 95197 86309 82059


Skew angle 120582cr



0∘ 120582cru 40010 40034 17956 18052 12624 10846120582crb 20002 20017 08980 09028 06312 05423

15∘ 120582cru 43946 44007 19716 19799 13859 11915120582crb 21154 21187 09517 09561 06683 05741

30∘ 120582cru 58966 59317 26496 26496 18586 16020120582crb 25365 25491 11519 11520 08047 06909


119873119900

119909119909cr 1198872

1205872

119863119898





0 005 01 015 02 025 03 035 040

2

4

6

8

10

12

14

Criti

cal b

uckl

ing

load

120582un

i

ra




119873119900

119909119909cr1198872

1205872




119888=

380GPa 120581119888=104WmK and 120572

119888= 74 times 10


119898= 70GPa 120581

119898= 204WmK and 120572

119898= 23 times








05 1 15 2 25500

1000

1500

2000

2500

3000

3500

LinearNonlinear

ab

Criti

cal b

uckl

ing

tem

pera

ture

ΔT

cr



0 5 10 15 20 25 301000

1500

2000

2500

3000

3500

4000

4500

Skew angle 120595


Criti

cal b

uckl

ing

tem

pera

ture

ΔT

cr




5 Conclusion



0 005 01 015 02 025 03 035 04500

1000

1500

2000

2500

3000

3500

Criti

cal b

uckl

ing

tem

pera

ture

ra

n = 0






Acknowledgment


References





























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












119875minus1

1198751

1198752

1198753

Si3N4119864 (Pa) 34843119890

9 00 minus3070119890minus4

2160119890minus7


120572 (1K) 58723119890minus6 00 9095119890

minus4 00 00

SUS304 119864 (Pa) 201041198909 00 3079119890

minus4


120572 (1K) 12330119890minus6 00 8086119890

minus4 00 00

002

004

006

008

01

012

014

016

018

Nor

mal

ized

cent

ral d

eflec

tion


ah

100 101 102 103 104


for Si3N4and 120588

119898= 8166 kgm3and 120581

119898= 1204WmK for


= 119879119898


119898=


a

b

r





2) The


2is 119864119888

= 151 GPa and for Alis 119864119898




119879119888 119879119898


300 300 Present 183570 110690 90260 85880[48] 183731 110288 90128 85870

400 300 Present 179778 107979 88626 83182[48] 179620 107860 87530 83090

600 300 Present 171205 101679 81253 76516[48] 171050 101550 81150 76420


[1205962

120588119888ℎ1198864

119863119888(1 minus ]2)]

14



0 005 01 015 02 025 03 035 045

10

15

20

25

30

35

40

Non

dim

ensio

naliz

ed m

ode 1

freq

uenc

y

ra

n = 0n = 5


119888= 400K 119879

119898=



120582cru =

1198730

119909119909cr1198872

1205872119863119888

120582crb =

1198730

119910119910cr1198872

1205872119863119888

(44)

05 1 15 2 251

2

3

4

5

6

7

ab

n = 0

n = 5

Criti

cal b

uckl

ing

load

120582cr

u



119888= 119864119888ℎ3




119879119888


300 [49] 176855 106681 96040 87113 82850Present 177122 106845 96188 87246 82976

400 [37] 174690 105174 94618 85738 81484Present 175488 105775 95197 86309 82059


Skew angle 120582cr



0∘ 120582cru 40010 40034 17956 18052 12624 10846120582crb 20002 20017 08980 09028 06312 05423

15∘ 120582cru 43946 44007 19716 19799 13859 11915120582crb 21154 21187 09517 09561 06683 05741

30∘ 120582cru 58966 59317 26496 26496 18586 16020120582crb 25365 25491 11519 11520 08047 06909


119873119900

119909119909cr 1198872

1205872

119863119898





0 005 01 015 02 025 03 035 040

2

4

6

8

10

12

14

Criti

cal b

uckl

ing

load

120582un

i

ra




119873119900

119909119909cr1198872

1205872




119888=

380GPa 120581119888=104WmK and 120572

119888= 74 times 10


119898= 70GPa 120581

119898= 204WmK and 120572

119898= 23 times








05 1 15 2 25500

1000

1500

2000

2500

3000

3500

LinearNonlinear

ab

Criti

cal b

uckl

ing

tem

pera

ture

ΔT

cr



0 5 10 15 20 25 301000

1500

2000

2500

3000

3500

4000

4500

Skew angle 120595


Criti

cal b

uckl

ing

tem

pera

ture

ΔT

cr




5 Conclusion



0 005 01 015 02 025 03 035 04500

1000

1500

2000

2500

3000

3500

Criti

cal b

uckl

ing

tem

pera

ture

ra

n = 0






Acknowledgment


References





























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119879119888 119879119898


300 300 Present 183570 110690 90260 85880[48] 183731 110288 90128 85870

400 300 Present 179778 107979 88626 83182[48] 179620 107860 87530 83090

600 300 Present 171205 101679 81253 76516[48] 171050 101550 81150 76420


[1205962

120588119888ℎ1198864

119863119888(1 minus ]2)]

14



0 005 01 015 02 025 03 035 045

10

15

20

25

30

35

40

Non

dim

ensio

naliz

ed m

ode 1

freq

uenc

y

ra

n = 0n = 5


119888= 400K 119879

119898=



120582cru =

1198730

119909119909cr1198872

1205872119863119888

120582crb =

1198730

119910119910cr1198872

1205872119863119888

(44)

05 1 15 2 251

2

3

4

5

6

7

ab

n = 0

n = 5

Criti

cal b

uckl

ing

load

120582cr

u



119888= 119864119888ℎ3




119879119888


300 [49] 176855 106681 96040 87113 82850Present 177122 106845 96188 87246 82976

400 [37] 174690 105174 94618 85738 81484Present 175488 105775 95197 86309 82059


Skew angle 120582cr



0∘ 120582cru 40010 40034 17956 18052 12624 10846120582crb 20002 20017 08980 09028 06312 05423

15∘ 120582cru 43946 44007 19716 19799 13859 11915120582crb 21154 21187 09517 09561 06683 05741

30∘ 120582cru 58966 59317 26496 26496 18586 16020120582crb 25365 25491 11519 11520 08047 06909


119873119900

119909119909cr 1198872

1205872

119863119898





0 005 01 015 02 025 03 035 040

2

4

6

8

10

12

14

Criti

cal b

uckl

ing

load

120582un

i

ra




119873119900

119909119909cr1198872

1205872




119888=

380GPa 120581119888=104WmK and 120572

119888= 74 times 10


119898= 70GPa 120581

119898= 204WmK and 120572

119898= 23 times








05 1 15 2 25500

1000

1500

2000

2500

3000

3500

LinearNonlinear

ab

Criti

cal b

uckl

ing

tem

pera

ture

ΔT

cr



0 5 10 15 20 25 301000

1500

2000

2500

3000

3500

4000

4500

Skew angle 120595


Criti

cal b

uckl

ing

tem

pera

ture

ΔT

cr




5 Conclusion



0 005 01 015 02 025 03 035 04500

1000

1500

2000

2500

3000

3500

Criti

cal b

uckl

ing

tem

pera

ture

ra

n = 0






Acknowledgment


References





























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119879119888


300 [49] 176855 106681 96040 87113 82850Present 177122 106845 96188 87246 82976

400 [37] 174690 105174 94618 85738 81484Present 175488 105775 95197 86309 82059


Skew angle 120582cr



0∘ 120582cru 40010 40034 17956 18052 12624 10846120582crb 20002 20017 08980 09028 06312 05423

15∘ 120582cru 43946 44007 19716 19799 13859 11915120582crb 21154 21187 09517 09561 06683 05741

30∘ 120582cru 58966 59317 26496 26496 18586 16020120582crb 25365 25491 11519 11520 08047 06909


119873119900

119909119909cr 1198872

1205872

119863119898





0 005 01 015 02 025 03 035 040

2

4

6

8

10

12

14

Criti

cal b

uckl

ing

load

120582un

i

ra




119873119900

119909119909cr1198872

1205872




119888=

380GPa 120581119888=104WmK and 120572

119888= 74 times 10


119898= 70GPa 120581

119898= 204WmK and 120572

119898= 23 times








05 1 15 2 25500

1000

1500

2000

2500

3000

3500

LinearNonlinear

ab

Criti

cal b

uckl

ing

tem

pera

ture

ΔT

cr



0 5 10 15 20 25 301000

1500

2000

2500

3000

3500

4000

4500

Skew angle 120595


Criti

cal b

uckl

ing

tem

pera

ture

ΔT

cr




5 Conclusion



0 005 01 015 02 025 03 035 04500

1000

1500

2000

2500

3000

3500

Criti

cal b

uckl

ing

tem

pera

ture

ra

n = 0






Acknowledgment


References





























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













05 1 15 2 25500

1000

1500

2000

2500

3000

3500

LinearNonlinear

ab

Criti

cal b

uckl

ing

tem

pera

ture

ΔT

cr



0 5 10 15 20 25 301000

1500

2000

2500

3000

3500

4000

4500

Skew angle 120595


Criti

cal b

uckl

ing

tem

pera

ture

ΔT

cr




5 Conclusion



0 005 01 015 02 025 03 035 04500

1000

1500

2000

2500

3000

3500

Criti

cal b

uckl

ing

tem

pera

ture

ra

n = 0






Acknowledgment


References





























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 005 01 015 02 025 03 035 04500

1000

1500

2000

2500

3000

3500

Criti

cal b

uckl

ing

tem

pera

ture

ra

n = 0






Acknowledgment


References





























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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